diff --git a/docs/plans/2026-07-30-issue-128-delivery-design.md b/docs/plans/2026-07-30-issue-128-delivery-design.md new file mode 100644 index 000000000..65215fff8 --- /dev/null +++ b/docs/plans/2026-07-30-issue-128-delivery-design.md @@ -0,0 +1,143 @@ +# Issue 128 Final Delivery Design + +**Date:** 2026-07-30 + +**Status:** Approved by the user (option A) + +**Target:** Existing pull request `QuantumBFS/quantum.harness#248` + +## Objective + +Finish Issue 128 as a reviewer-ready, reproducible package. Preserve the +proved `11791/2911 = 4.050498110614909...x` resource improvement as the +official result, add a polished technical report and machine-readable +summaries, and report the strict fivefold investigation as an honest +feasibility appendix rather than a completed certificate. + +## Frozen Scientific Claim + +The benchmark is the periodic `12 x 12` spin-1/2 isotropic Heisenberg model + +```text +H = sum_ (X_i X_j + Y_i Y_j + Z_i Z_j) / 4, +T = 1, operator-norm tolerance = 1e-6. +``` + +The pinned published rigorous baseline uses 393 Trotter steps and 11,791 +merged group exponentials. The submitted certificate uses 97 steps and 2,911 +merged group exponentials. The 97-step exact-rational error bound is below +`1e-6`; the 96-step bound is above it. The proof, circuit, normalization, +tolerance, resource model, main certificate, and D4 sidecar are immutable. + +## Chosen Approach + +Extend the existing proof branch with four reviewer-facing layers: + +1. a standard LaTeX article of approximately 6--8 pages; +2. compact JSON and plain-text result summaries; +3. a verifier transcript and SHA-256 manifest; and +4. a precise update to the existing PR description. + +The exact D5 artifact is follow-up evidence only. No `5x` result may be stated +or implied. The unresolved translation-coupled D4 and explicit D8/tail proof +gates, and the absence of an accepted 78-step global certificate, must be +visible wherever the conditional fivefold arithmetic appears. + +## Deliverables + +All final files live under +`tracks/qcs/solutions/WangTheoPhys/issue128`: + +```text +docs/report/issue128-technical-report.tex +docs/report/references.bib +docs/report/output/issue128-technical-report.pdf +artifacts/issue128-summary.json +artifacts/issue128-summary.txt +artifacts/verification-transcript.txt +artifacts/SHA256SUMS +``` + +The PDF is versioned because it is a competition deliverable. Auxiliary LaTeX +files and rendered page images are temporary and are not committed. + +## Report Structure + +No challenge template exists, so the report uses the standard `article` class. +It contains: + +1. abstract and contribution summary; +2. benchmark and resource accounting; +3. reconstruction of the published rigorous baseline; +4. local logarithm, Lie projection, concrete Pauli representation, and exact + anticommuting-group method; +5. finite-step right-generator error ledger; +6. verifier architecture and small-size dense cross-check; +7. certified result and exact integer boundary; +8. strict fivefold feasibility evidence and unresolved gates; and +9. limitations and copy-paste reproduction commands. + +External citations are restricted to verified primary sources. + +## Trust Boundary and Data Flow + +The certificate and sidecars remain the source of truth. A packaging script +checks their frozen hashes, projects exact fields into JSON and text, runs the +fast main verifier and exact D5 verifier, and records their complete output. +The final manifest binds the PDF, summaries, transcript, certificates, and +report sources. + +- integers, `Fraction`, rational intervals, and symplectic Pauli algebra are + trusted; +- heuristic grouping is untrusted discovery and is rechecked exactly; +- floating point is presentation-only or part of the dense sanity check; +- PDF and summaries explain the certificate but never replace it. + +## Fivefold Reporting Policy + +At 78 steps the unchanged compilation rule gives 2,341 groups and the +conditional ratio `11791/2341 = 5.036736...`. This arithmetic does not prove a +78-step error bound. The D5 sidecar has 605,832 exact terms, 123,106 +same-support pairwise-anticommuting groups, and site-density upper bound +`11.23706750025696`; it narrows the gap without closing D4 or D8/tail. + +Every conditional fivefold statement must be accompanied by: + +```text +No 78-step global error certificate is claimed or supplied. +``` + +## Validation + +Release requires all of the following: + +- the full default Issue 128 test suite; +- the main fast verifier and exact D5 verifier; +- the 97-accept/96-reject boundary checks; +- frozen main-certificate and D4 hashes; +- deterministic JSON/text regeneration; +- SHA-256 verification of every manifest entry; +- LaTeX compilation without fatal errors, undefined references, or missing + citations; +- PDF text/page checks and visual inspection of every rendered page; +- an unsupported-fivefold-language scan; +- an explicit Git allowlist and remote-head drift check. + +If a follow-up check fails, remove its affirmative claim; never weaken the +official 4.050x proof or substitute a numerical estimate. + +## Git and PR Strategy + +Work only in the isolated Issue 128 clone. Explicitly stage approved files; +never use `git add -A`. Push to the existing fork head branch +`JunkaiWang-TheoPhy:codex/issue-128-trotter-certificate` only if its remote head +is still an ancestor of the audited local head. Do not force-push and do not +open a competing PR. Update PR #248 with artifact links, exact results, +validation commands, and the fivefold non-claim. + +## Definition of Done + +The report and data package are committed, every validation gate passes, the +audited history is fast-forward pushed to the existing PR branch, the remote +SHA equals the local SHA, and PR #248 exposes accurate reproduction steps and +limitations. Maintainer review and merge are external outcomes. diff --git a/hpc/issue128_d8_array.sbatch b/hpc/issue128_d8_array.sbatch new file mode 100755 index 000000000..a84c19a8c --- /dev/null +++ b/hpc/issue128_d8_array.sbatch @@ -0,0 +1,38 @@ +#!/usr/bin/env bash +#SBATCH --job-name=issue128-d8 +#SBATCH --partition=xhacnormalb +#SBATCH --account=giggleliu +#SBATCH --qos=user_acamtw70yu +#SBATCH --nodes=1 +#SBATCH --ntasks=1 +#SBATCH --cpus-per-task=17 +#SBATCH --mem=64G +#SBATCH --time=04:30:00 +#SBATCH --array=0-30%16 +#SBATCH --output=results/slurm-%x-%A_%a.out +#SBATCH --error=results/slurm-%x-%A_%a.err + +set -euo pipefail + +: "${ISSUE128_RUN_ID:?ISSUE128_RUN_ID must name the result run}" +ISSUE128_ORDER="${ISSUE128_ORDER:-8}" + +export PYTHONHASHSEED=0 +export OMP_NUM_THREADS=1 +export OPENBLAS_NUM_THREADS=1 +export MKL_NUM_THREADS=1 +export NUMEXPR_NUM_THREADS=1 + +REPOSITORY_ROOT="${SLURM_SUBMIT_DIR:?}" +ISSUE_ROOT="$REPOSITORY_ROOT/tracks/qcs/solutions/WangTheoPhys/issue128" +export PYTHONPATH="$ISSUE_ROOT/src${PYTHONPATH:+:$PYTHONPATH}" +CELL_ID=$(printf "stage-%02d" "$SLURM_ARRAY_TASK_ID") +CELL_DIR="$REPOSITORY_ROOT/results/$ISSUE128_RUN_ID/cells/$CELL_ID" +mkdir -p "$CELL_DIR" + +cd "$ISSUE_ROOT" +"$REPOSITORY_ROOT/.venv/bin/python" -u scripts/build_d8_shard.py \ + --stage-index "$SLURM_ARRAY_TASK_ID" \ + --order "$ISSUE128_ORDER" \ + --output "$CELL_DIR/shard.json.gz" \ + --manifest "$CELL_DIR/manifest.json" diff --git a/hpc/issue128_d8_reduce.sbatch b/hpc/issue128_d8_reduce.sbatch new file mode 100755 index 000000000..55db8cb0d --- /dev/null +++ b/hpc/issue128_d8_reduce.sbatch @@ -0,0 +1,34 @@ +#!/usr/bin/env bash +#SBATCH --job-name=issue128-d8-reduce +#SBATCH --partition=xhacnormalb +#SBATCH --account=giggleliu +#SBATCH --qos=user_acamtw70yu +#SBATCH --nodes=1 +#SBATCH --ntasks=1 +#SBATCH --cpus-per-task=26 +#SBATCH --mem=96G +#SBATCH --time=01:15:00 +#SBATCH --output=results/slurm-%x-%j.out +#SBATCH --error=results/slurm-%x-%j.err + +set -euo pipefail +: "${ISSUE128_RUN_ID:?ISSUE128_RUN_ID must name the result run}" +ISSUE128_ORDER="${ISSUE128_ORDER:-8}" + +export PYTHONHASHSEED=0 +export OMP_NUM_THREADS=1 +export OPENBLAS_NUM_THREADS=1 +export MKL_NUM_THREADS=1 + +REPOSITORY_ROOT="${SLURM_SUBMIT_DIR:?}" +ISSUE_ROOT="$REPOSITORY_ROOT/tracks/qcs/solutions/WangTheoPhys/issue128" +export PYTHONPATH="$ISSUE_ROOT/src${PYTHONPATH:+:$PYTHONPATH}" +RUN_ROOT="$REPOSITORY_ROOT/results/$ISSUE128_RUN_ID" +mkdir -p "$RUN_ROOT/reduced" +cd "$ISSUE_ROOT" +"$REPOSITORY_ROOT/.venv/bin/python" -u scripts/reduce_d8_shards.py \ + --input-root "$RUN_ROOT/cells" \ + --expected-stages 31 \ + --order "$ISSUE128_ORDER" \ + --output "$RUN_ROOT/reduced/degree-${ISSUE128_ORDER}.json.gz" \ + --summary "$RUN_ROOT/reduced/summary.json" diff --git a/hpc/issue128_screen_array.sbatch b/hpc/issue128_screen_array.sbatch new file mode 100755 index 000000000..b4fa5b048 --- /dev/null +++ b/hpc/issue128_screen_array.sbatch @@ -0,0 +1,35 @@ +#!/usr/bin/env bash +#SBATCH --job-name=issue128-screen +#SBATCH --partition=xhacnormalb +#SBATCH --account=giggleliu +#SBATCH --qos=user_acamtw70yu +#SBATCH --nodes=1 +#SBATCH --ntasks=1 +#SBATCH --cpus-per-task=5 +#SBATCH --mem=16G +#SBATCH --time=02:00:00 +#SBATCH --array=0-23%24 +#SBATCH --output=results/slurm-%x-%A_%a.out +#SBATCH --error=results/slurm-%x-%A_%a.err + +set -euo pipefail +: "${ISSUE128_RUN_ID:?ISSUE128_RUN_ID must name the result run}" +ISSUE128_SCREEN_ORDER="${ISSUE128_SCREEN_ORDER:-6}" + +export PYTHONHASHSEED=0 +export OMP_NUM_THREADS=1 +export OPENBLAS_NUM_THREADS=1 +export MKL_NUM_THREADS=1 + +REPOSITORY_ROOT="${SLURM_SUBMIT_DIR:?}" +ISSUE_ROOT="$REPOSITORY_ROOT/tracks/qcs/solutions/WangTheoPhys/issue128" +export PYTHONPATH="$ISSUE_ROOT/src${PYTHONPATH:+:$PYTHONPATH}" +CELL_ID=$(printf "permutation-%02d" "$SLURM_ARRAY_TASK_ID") +CELL_DIR="$REPOSITORY_ROOT/results/$ISSUE128_RUN_ID/screen/$CELL_ID" +mkdir -p "$CELL_DIR" +cd "$ISSUE_ROOT" +"$REPOSITORY_ROOT/.venv/bin/python" -u scripts/screen_matching_order.py \ + --permutation-index "$SLURM_ARRAY_TASK_ID" \ + --order "$ISSUE128_SCREEN_ORDER" \ + --output "$CELL_DIR/result.json" \ + --manifest "$CELL_DIR/manifest.json" diff --git a/hpc/issue128_verify.sbatch b/hpc/issue128_verify.sbatch new file mode 100755 index 000000000..654ce7789 --- /dev/null +++ b/hpc/issue128_verify.sbatch @@ -0,0 +1,66 @@ +#!/usr/bin/env bash +#SBATCH --job-name=issue128-verify +#SBATCH --partition=xhacnormalb +#SBATCH --account=giggleliu +#SBATCH --qos=user_acamtw70yu +#SBATCH --nodes=1 +#SBATCH --ntasks=1 +#SBATCH --cpus-per-task=9 +#SBATCH --mem=32G +#SBATCH --time=04:30:00 +#SBATCH --output=results/slurm-%x-%j.out +#SBATCH --error=results/slurm-%x-%j.err + +set -euo pipefail +: "${ISSUE128_RUN_ID:?ISSUE128_RUN_ID must name the result run}" + +export PYTHONHASHSEED=0 +export OMP_NUM_THREADS=1 +export OPENBLAS_NUM_THREADS=1 +export MKL_NUM_THREADS=1 + +REPOSITORY_ROOT="${SLURM_SUBMIT_DIR:?}" +ISSUE_ROOT="$REPOSITORY_ROOT/tracks/qcs/solutions/WangTheoPhys/issue128" +export PYTHONPATH="$ISSUE_ROOT/src${PYTHONPATH:+:$PYTHONPATH}" +OUT="$REPOSITORY_ROOT/results/$ISSUE128_RUN_ID/verification" +mkdir -p "$OUT" +cd "$ISSUE_ROOT" + +set +e +"$REPOSITORY_ROOT/.venv/bin/python" -u scripts/verify.py \ + certificates/issue128-certificate.json --deep \ + >"$OUT/deep-verifier.stdout" 2>"$OUT/deep-verifier.stderr" +VERIFY_RC=$? +"$REPOSITORY_ROOT/.venv/bin/python" -m pytest -q -o addopts='' \ + >"$OUT/pytest.stdout" 2>"$OUT/pytest.stderr" +PYTEST_RC=$? +set -e + +"$REPOSITORY_ROOT/.venv/bin/python" - "$OUT" "$VERIFY_RC" "$PYTEST_RC" <<'PY' +import datetime +import hashlib +import json +import pathlib +import sys + +root = pathlib.Path(sys.argv[1]) +verify_rc = int(sys.argv[2]) +pytest_rc = int(sys.argv[3]) +files = {} +for path in sorted(root.glob("*.std*")): + files[path.name] = hashlib.sha256(path.read_bytes()).hexdigest() +payload = { + "schema_version": 1, + "kind": "issue128_hpc_verification", + "status": "complete" if verify_rc == 0 and pytest_rc == 0 else "failed", + "deep_verifier_exit_code": verify_rc, + "pytest_exit_code": pytest_rc, + "completed_at": datetime.datetime.now(datetime.timezone.utc).isoformat(), + "sha256": files, +} +(root / "manifest.json.tmp").write_text(json.dumps(payload, sort_keys=True, indent=2) + "\n") +(root / "manifest.json.tmp").replace(root / "manifest.json") +PY + +test "$VERIFY_RC" -eq 0 +test "$PYTEST_RC" -eq 0 diff --git a/scripts/harness_slurm.sh b/scripts/harness_slurm.sh index d14bc7f7e..bb816840b 100755 --- a/scripts/harness_slurm.sh +++ b/scripts/harness_slurm.sh @@ -21,6 +21,7 @@ # precheck resolve profile, test ssh, capture git dirty status # probe-partitions ssh sinfo and print a parsed candidate table # submit ... build+run sbatch, or validate with --test-only +# (repeat --env KEY=VALUE for explicit job metadata) # status squeue the job, parse state + pending-reason category # fetch rsync results// back from the cluster # classify sacct + per-cell manifests -> cell outcome table @@ -208,6 +209,7 @@ cmd_probe_partitions() { cmd_submit() { local array="" run_spec="" command="" entrypoint="" partition="" walltime="" cpus="" \ script="scripts/harness_array_sbatch.sh" extra="" test_only="false" + local -a job_env=() while [[ $# -gt 0 ]]; do case "$1" in --array) array="$2"; shift 2 ;; @@ -218,6 +220,7 @@ cmd_submit() { --time) walltime="$2"; shift 2 ;; --cpus) cpus="$2"; shift 2 ;; --script) script="$2"; shift 2 ;; + --env) job_env+=("$2"); shift 2 ;; --extra) extra="$2"; shift 2 ;; --test-only) test_only="true"; shift ;; *) die "submit: unknown flag $1" ;; @@ -237,6 +240,12 @@ cmd_submit() { elif [[ -n "$array" ]]; then die "submit: --array requires --run-spec; use a plain --script for single jobs" fi + local assignment + for assignment in "${job_env[@]}"; do + [[ "$assignment" =~ ^[A-Za-z_][A-Za-z0-9_]*=[^,[:space:]]+$ ]] || \ + die "submit: --env must be KEY=VALUE without commas or whitespace" + exports="${exports},${assignment}" + done local extra_partition="" extra_gres="" extra_status=0 \ extra_partition_set="false" extra_gres_set="false" diff --git a/skills/using-slurm/profiles/xh5-acamtw.toml b/skills/using-slurm/profiles/xh5-acamtw.toml new file mode 100644 index 000000000..9183e7fec --- /dev/null +++ b/skills/using-slurm/profiles/xh5-acamtw.toml @@ -0,0 +1,62 @@ +[identity] +name = "xh5-acamtw" +purpose = "Issue 128 exact-rational CPU sprint on hpccube Xiongheng" +maintainer = "Wander" + +[connection] +repo_path_remote = "/work/home/acamtw70yu/quantum-harness-issue128-hpc" +login_shell = false + +[connection.ssh] +alias = "xh5-acamtw" +host = "xh5.hpccube.com" +user = "acamtw70yu" +port = 65061 + +[scheduler] +type = "slurm" +default_partition = "xhacnormalb" + +[[partitions]] +name = "xhacnormalb" +class = "default-cpu" +cores = 128 +memory = "513500M" +max_wall = "333-08:00:00" +gpu = "" + +[filesystem] +home = "/work/home/acamtw70yu" +scratch = "" +project = "/work/home/acamtw70yu/quantum-harness-issue128-hpc" +quota = "" + +[network] +internet_from_login = true +internet_from_compute = false + +[region] +region = "mainland_china" + +[limits.hard] +max_walltime = "06:00:00" +max_nodes = 1 +max_cpus = 64 +max_array_size = 200 + +[limits.soft] +warn_walltime = "04:30:00" +warn_cpus = 32 +unusual_partitions = [] + +[limits.paths] +allowed_roots = ["/work/home/acamtw70yu/quantum-harness-issue128-hpc/results"] + +[commands] +squeue = "squeue -u $USER" +sacct = "sacct --format=JobID,State,ExitCode,MaxRSS,Elapsed" +sinfo = "sinfo -o '%P %a %.10l %.6D %.6t'" +quota_command = "sacctmgr -n -P show assoc where user=$USER format=Account,Partition,QOS,GrpTRES,MaxTRES" + +[notes] +text = "Live-probed 2026-07-31: account giggleliu, QOS user_acamtw70yu, CPU partition xhacnormalb, 128 cores and about 501 GiB per node. GPU partitions are intentionally excluded because the sprint is exact Python/Fraction work." diff --git a/skills/using-slurm/profiles/xh5-cfys.toml b/skills/using-slurm/profiles/xh5-cfys.toml new file mode 100644 index 000000000..d8586c6b1 --- /dev/null +++ b/skills/using-slurm/profiles/xh5-cfys.toml @@ -0,0 +1,56 @@ +[identity] +name = "xh5-cfys" +purpose = "Secondary Issue 128 discovery and independent verification on hpccube Xiongheng" +maintainer = "Wander" + +[connection] +repo_path_remote = "/work/home/cfys01/quantum-harness-issue128-hpc" +login_shell = false + +[connection.ssh] +alias = "hpccube-xh5" +host = "xh5.hpccube.com" +user = "cfys01" +port = 65061 + +[scheduler] +type = "slurm" +default_partition = "xhacnormalb" + +[[partitions]] +name = "xhacnormalb" +class = "default-cpu" +cores = 128 +memory = "513500M" +max_wall = "333-08:00:00" +gpu = "" + +[limits.hard] +max_walltime = "06:00:00" +max_nodes = 1 +max_cpus = 32 +max_array_size = 100 + +[limits.soft] +warn_walltime = "04:30:00" +warn_cpus = 16 +unusual_partitions = [] + +[limits.paths] +allowed_roots = ["/work/home/cfys01/quantum-harness-issue128-hpc/results"] + +[network] +internet_from_login = true +internet_from_compute = false + +[region] +region = "mainland_china" + +[commands] +squeue = "squeue -u $USER" +sacct = "sacct --format=JobID,State,ExitCode,MaxRSS,Elapsed" +sinfo = "sinfo -o '%P %a %.10l %.6D %.6t'" +quota_command = "sacctmgr -n -P show assoc where user=$USER format=Account,Partition,QOS,GrpTRES,MaxTRES" + +[notes] +text = "Secondary account already had 36 running jobs at the 2026-07-31 probe. Do not use it to bypass shared account/QOS limits." diff --git a/tracks/qcs/solutions/WangTheoPhys/README.md b/tracks/qcs/solutions/WangTheoPhys/README.md new file mode 100644 index 000000000..195496419 --- /dev/null +++ b/tracks/qcs/solutions/WangTheoPhys/README.md @@ -0,0 +1,268 @@ +# 🔭 WangTheoPhys: machine-certified local-logarithm Trotter bounds + +## Team + +| | | +|---|---| +| **Team name** | WangTheoPhys | +| **Members** | Junkai Wang, WangTheoPhys@outlook.com | +| **Challenge** | [#128 — Tighter provable Trotter error bounds for a concrete 2D Hamiltonian](https://github.com/QuantumBFS/quantum.harness/issues/128) | +| **Track** | `qcs` — selected from the issue's `Method: Quantum Circuit Simulation` field | + +## Result + +For the periodic spin-1/2 square-lattice Heisenberg Hamiltonian + +```text +H = Σ_ (X_i X_j + Y_i Y_j + Z_i Z_j) / 4 +L = 12, N = 144, T = 1, operator-norm tolerance = 1e-6 +``` + +we certify the same five-copy fourth-order Suzuki circuit with 97 Trotter +steps, compared with 393 steps from the pinned published +Childs–Su–Tran–Wiebe–Zhu high-order commutator-bound instantiation. + +| Resource | Published rigorous baseline | This certificate | +|---|---:|---:| +| Trotter steps | 393 | 97 | +| Merged group exponentials | 11,791 | 2,911 | +| Bond propagators | 848,952 | 209,592 | +| CNOT upper bound (3 per bond) | 2,546,856 | 628,776 | + +The exact resource improvement is + +```text +11791 / 2911 = 4.050498110614909... +``` + +At 97 steps the certified global operator-norm error is +`9.958938494314325e-7`. At 96 steps the bound is +`1.050565873970784e-6`, so 97 is the smallest integer accepted by this +certificate. + +## What is new + +The circuit, fragmentation, and cost model are identical on both sides. The +gain comes entirely from changing where norm inequalities enter the proof. + +The published baseline expands partial Hamiltonian sums into individual +degree-five nested commutators and takes local Pauli-l1 norms term by term. +Our proof instead: + +1. computes the homogeneous free-associative-word logarithm of the complete + fourth-order formula; +2. applies the Dynkin–Specht–Wever projection to recover the degree-five and + degree-seven Lie elements; +3. maps those Lie elements into the concrete four-matching Heisenberg + representation; +4. combines identical local Pauli strings before taking a norm bound; +5. partitions the leading right-generator Pauli terms into exactly checked + pairwise-anticommuting groups; +6. converts the logarithm into the exact right generator; and +7. closes every degree at finite step size with a rational geometric locality + tail. + +For a group \(K_g=\sum_j c_jP_j\) whose Pauli strings anticommute pairwise, + +```text +||K_g|| <= sqrt(Σ_j upper(|c_j|)^2). +``` + +The submitted sidecar covers all 75,324 canonical leading-defect terms +exactly once with 7,576 groups of size at most 10. Every pairwise +anticommutation relation and every outward-rounded rational square-root bound +is checked independently. The untrusted greedy search chooses the partition +only; it is not part of the trusted proof. This tightens the leading +per-cell bound from `20.160968407335066` to `6.472926505087888`, a factor of +`3.114660484942633`. + +A further exact local lemma tightens all higher-degree growth estimates. For +one Heisenberg bond `h=(XX+YY+ZZ)/4`, every phase-free two-qubit Pauli string +anticommutes with either zero or exactly two of `XX`, `YY`, and `ZZ`. +Consequently + +```text +||[h,P]||_Pauli-l1 <= ||P||_Pauli-l1, +``` + +so the bond commutator growth constant is `1`, not the generic triangle bound +`3/2`. The verifier checks all 16 local Pauli cases exactly. + +Writing + +```text +log S4(t) = t A + t^5 B + t^7 C + O(t^9), +``` + +the right logarithmic generator obeys + +```text +S4'(t) S4(t)^(-1) - A + = 5 t^4 B + + 2 t^5 [A,B] + + t^6 (7 C + 2/3 [A,[A,B]]) + + t^7 (3 [A,C] + 1/6 [A,[A,[A,B]]]) + + R_{>=8}(t). +``` + +The remainder is not discarded. Local support growth, conjugation Taylor +factorials, and the Duhamel integration factor reduce it to an explicitly +summable rational geometric series. + +## Error ledger at 97 steps + +| Right-generator degree | Global contribution | +|---:|---:| +| 4 | `5.264367936822258e-7` | +| 5 | `1.3834875367446118e-7` | +| 6 | `1.579057000594835e-7` | +| 7 | `1.8946570806914694e-8` | +| 8 and above | `1.5425603120834724e-7` | +| **Total** | **`9.958938494314325e-7`** | + +All certificate values are stored as exact rational numerators and +denominators in +[`issue128/certificates/issue128-certificate.json`](issue128/certificates/issue128-certificate.json). +The large exact partition is stored separately in +[`issue128/certificates/issue128-d4-groups.json`](issue128/certificates/issue128-d4-groups.json) +and bound to the main certificate by SHA-256. + +## Published baseline control + +The baseline instantiates the higher-order commutator theorem of: + +- A. M. Childs, Y. Su, M. C. Tran, N. Wiebe, and S. Zhu, + [*A Theory of Trotter Error*](https://arxiv.org/abs/1912.08854); +- A. Schubert and C. B. Mendl, + [*Trotter error with commutator scaling for the Fermi-Hubbard model*](https://arxiv.org/abs/2306.10603). + +The verifier expands partial sums before applying local Pauli-l1 bounds and +checks all coefficients with rational intervals. All 31 allowed theorem +centers were scanned; center 20 gives the smallest published-baseline +instantiation, with site-density upper bound `164.97591695274392` and 393 +steps. + +We separately audited the standard recursive Suzuki orders 2, 4, 6, and 8. +The authoritative schema-v3 certificate records 179,785, 24,241, 55,501, and +251,251 merged groups, respectively. These are contextual audit values and do +not enter the primary 393-versus-97 comparison. An earlier short report copied +a legacy set of auxiliary values; the discrepancy and its non-impact on the +main claim are documented in the [report erratum](issue128/docs/report/ERRATA.md). + +Optimized/processed formulas ranked empirically on random matrices are not +treated as deterministic worst-case bounds for this instance. Adapting their +coefficients to an interval-certified comparison is a useful follow-up, but +their empirical constants cannot replace the pinned rigorous baseline. + +## Independent checks and negative results + +- A fast verifier checks the sidecar digest, exact term coverage, every + symplectic anticommutation relation, every rational group bound, the + integer step boundary, and all resource arithmetic. +- A deep verifier additionally regenerates the published baseline, all 75,324 + canonical leading-defect coefficients, the degree-seven majorant, and the + finite tail directly from the formula. +- A degenerate 2x2 periodic-algebra sanity check gives actual spectral-norm + error `9.128277711816321e-10`, below its outward-rounded certificate + `2.7666666666666666e-8`. +- The degree-three Lie representation of the four matching fragments has + exact rank 20, equal to the free-Lie dimension `(4^3-4)/3 = 20`. Thus the + current fragmentation has no free model-specific reduction in fourth-order + order conditions. +- A three-site SU(2)-cluster decomposition improves local block counts but not + the conservative compiled CNOT bound once the three-site propagator cost is + included. +- A [quantitative 5×/10× follow-up audit](issue128/docs/followups/2026-07-29-5x-10x-audit.md) + shows that D4 tightening alone cannot reach the next threshold: 5× also + requires tighter D5-D10/tail certificates, while 10× requires an effective + order lift such as a certified processor or shorter sixth-order kernel. + +## Final delivery package + +The reviewer-facing package freezes the certified `4.050498110614909×` +result and binds the human-readable report to the exact proof artifacts: + +- [technical report PDF](issue128/docs/report/output/issue128-technical-report.pdf) + and [LaTeX source](issue128/docs/report/issue128-technical-report.tex); +- [machine-readable JSON summary](issue128/artifacts/issue128-summary.json) and + [plain-text result summary](issue128/artifacts/issue128-summary.txt); +- [verification transcript](issue128/artifacts/verification-transcript.txt); +- [SHA-256 manifest](issue128/artifacts/SHA256SUMS) covering the report, + summaries, transcript, certificates, and exact sidecars. + +The following derived materials sit outside that frozen ten-file manifest and +must be read as explanations or follow-up engineering, not as replacements for +the exact certificate: + +| Material | Purpose | Claim status | +|---|---|---| +| [30-page research manuscript](issue128/docs/manuscript/output/pdf/issue128-proof-carrying-trotter-paper.pdf) | Full derivation, proofs, literature positioning, limitations, and appendices | Explains the frozen 4.050498× certificate | +| [Manuscript source and build instructions](issue128/docs/manuscript/README.md) | Rebuild the PDF and all four vector figures | Derived, reproducible document | +| [Citation audit](issue128/docs/manuscript/CITATION_AUDIT.md) | Validate all 23 references by DOI, arXiv ID, or official catalog | Bibliographic QA | +| [10-minute talk script](issue128/docs/presentation/issue128-10-minute-talk.md) | Eight-slide presentation and Q&A boundaries | Reports only the certified 4.050498× result | +| [Six-hour HPC log](issue128/docs/report/issue128-hpc-sprint-log.md) | Exact D8 shard/reducer follow-up | Exploratory; no fivefold certificate | +| [Short-report erratum](issue128/docs/report/ERRATA.md) | Correct a legacy auxiliary recursive-order table | Main result unchanged | + +The fivefold follow-up includes an independently checked degree-five sidecar: +605,832 canonical Pauli terms, 123,106 same-support pairwise-anticommuting +groups, and site-density upper bound `11.23706750025696`. Conditional resource +arithmetic at `r=78` would be `11791/2341 = 5.036736437419906...`, but the D4 +budget and D8-plus global tail gates are not closed. **No 78-step global error +certificate is claimed or supplied.** + +## Reproduce + +From this directory: + +```bash +python -m pip install -e '.[test]' +pytest -q +PYTHONPATH=src python scripts/verify.py \ + certificates/issue128-certificate.json +PYTHONPATH=src python scripts/build_d5_certificate.py --verify-only +PYTHONPATH=src python scripts/package_delivery.py --check +shasum -a 256 -c artifacts/SHA256SUMS +``` + +The fast verifier checks the submitted exact-rational certificate. The deep +mode regenerates the expensive local algebra: + +```bash +PYTHONPATH=src python scripts/verify.py \ + certificates/issue128-certificate.json --deep +``` + +Regenerate the certificate: + +```bash +PYTHONPATH=src python scripts/build_v3_certificate.py +``` + +Build and validate the long-form paper: + +```bash +cd docs/manuscript +make pdf +``` + +The default test suite excludes explicitly marked research-reproduction tests. +The deep verifier independently regenerates the expensive local algebra. + +## Files + +```text +issue128/ +├── certificates/ exact certificate and verification summaries +├── artifacts/ reviewer summaries, transcript, and frozen manifest +├── docs/manuscript/ 30-page paper, source, figures, and citation audit +├── docs/presentation/ 10-minute presentation script +├── docs/report/ short report, HPC log, and erratum +├── scripts/ certificate builder, verifier, dense cross-check +├── src/trottercert/ exact Pauli/Lie/interval/locality implementation +├── tests/ fast and marked slow regression tests +└── pyproject.toml +``` + +The trusted verification path uses integers, `Fraction`, rational interval +arithmetic, exact symplectic Pauli multiplication, and outward rounding. +Floating point is used only for discovery and dense sanity checks. diff --git a/tracks/qcs/solutions/WangTheoPhys/issue128/artifacts/SHA256SUMS b/tracks/qcs/solutions/WangTheoPhys/issue128/artifacts/SHA256SUMS new file mode 100644 index 000000000..312761b84 --- /dev/null +++ b/tracks/qcs/solutions/WangTheoPhys/issue128/artifacts/SHA256SUMS @@ -0,0 +1,10 @@ +c623999ecac8a7718ed0971a497ece8148e4d70a293d5ed90338db7cea9d735b artifacts/issue128-summary.json +77228cdb866363c2c8efebc456b6f4c7591ec2f515ebdfd9090cd8a029976c8a artifacts/issue128-summary.txt +f4c9e2c4850052ddecfe2f8a8b584cdcf16eddc5eb0b86fffcb5bd93a237db73 artifacts/verification-transcript.txt +0a09623ce3b292a3637065c870fb3153bbdcddce30aef968565c4db3ddfc7201 certificates/issue128-certificate.json +a397414bb0229fb1ebdb38798aa781fb89dbb9d5cdbed94c7cd2e9120da62718 certificates/issue128-d4-groups.json +c5e8968a93b4497b41fe42c0e364324388272e183e1fcd20a536bc988f5361dd certificates/issue128-d5-groups.json.gz +ffeba2a7d2f880a14b7ee1600740885d46d702f70de1c94dfb6476efe785a4e8 certificates/issue128-small-crosscheck.json +fc963a624ab29a301df4be06de352b33b0fb531ef9b4196e298fcc61ccbff19e docs/report/issue128-technical-report.tex +16b2ef450a2ffd7662c4bd56a244f709f6dc09173175e2fc1e2a3bca7f5ea0f8 docs/report/references.bib +f50090a26979e0ba1bda7fe74a8a2673b101bc2d7115bda939e69953e14f4ba8 docs/report/output/issue128-technical-report.pdf diff --git a/tracks/qcs/solutions/WangTheoPhys/issue128/artifacts/issue128-summary.json b/tracks/qcs/solutions/WangTheoPhys/issue128/artifacts/issue128-summary.json new file mode 100644 index 000000000..e9a32468c --- /dev/null +++ b/tracks/qcs/solutions/WangTheoPhys/issue128/artifacts/issue128-summary.json @@ -0,0 +1,176 @@ +{ + "benchmark": { + "formula": "five_copy_suzuki_fourth_order", + "formula_order": 4, + "length": 12, + "model": "periodic_square_spin_half_isotropic_heisenberg", + "normalization": "(XX+YY+ZZ)/4", + "sites": 144, + "stage_count": 31, + "time": [ + 1, + 1 + ], + "tolerance": [ + 1, + 1000000 + ] + }, + "certified_result": { + "accepted_at_97": true, + "bond_propagators": 209592, + "cnot_upper": 628776, + "error_upper": [ + 599670422745656108926597685630153943951201095508880641428475695866175124754901808839128699565476683604156496265888897758053359535162204152523497215954848848755961894315360438774573129074241941927331478665448272993350597926513476349231331606355445437094033812134866153343154999415560759372473990440779968348417384816501377794329390792937346394068204978527979293780943439329901747152018428695684650908057947950752275565153549211215094936727050696808115471285220887575343320538864415440435484823724401, + 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18628098815762331308955745375744031451312712544003951915143323587191765140913687282590556923012229284286727492973637946908091979520580506941522631107558919299568456616026386835101726158947303895942669287179491573040382549259481731100862869066053794714470474282725565473839414726457780840434975235932140103612890697147056306440018341055946734378261970229499903599140373782008230578526026304631306088724623197505724600262101991677390833449941204813309538996425837404480621025826120161090811331145602190211, + 17731490501737330132971023158265461256557754955863122336739410011121346612799210695034681326844820557504793235783494060750359613343514225496752334858555018334128350507379874818501208678928093800230834997231437643227318036143974224856394241661177886302136098849703389670867925933947203616654322290404728008558748552082205606402627773932681885052986146827660046233143704985663445575306347781684791090345692853436416000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 + ], + "previous_step_error_upper_decimal_outward": "1.050565873970784e-6", + "rejected_at_96": true, + "status": "certified", + "steps": 97 + }, + "error_ledger": { + "degree4": { + "decimal_outward": "5.264367936822258e-7", + "exact": [ + 58256338545790990093628258829867, + 110661601250000000000000000000000000000 + ] + }, + "degree5": { + "decimal_outward": "1.383487536744612e-7", + "exact": [ + 5940239109672385481139, + 42936701285000000000000000000 + ] + }, + "degree6": { + "decimal_outward": "1.579057000594836e-7", + "exact": [ + 1841434385955714104548091490621189043370256643471762447183931887538056269601500787798709217, + 11661608069006000000000000000000000000000000000000000000000000000000000000000000000000000000000000 + ] + }, + "degree7": { + "decimal_outward": "1.894657080691470e-8", + "exact": [ + 5357976462796315520172382471863567130110769930415287341551795662614168808804502363396127651, + 282793995673395500000000000000000000000000000000000000000000000000000000000000000000000000000000000 + ] + }, + "tail": { + "decimal_outward": "1.542560312083473e-7", + "exact": [ + 92884175857284035685491336120458886544591778856897518237878840530272528365781587849919976832473665497571875376721654086499568100219300178381272091488081075497432947804603406574221283511468320089988532596441734375778223324921737782346334828088922482572379748281546586565691964441068690267790066150715130700988982046174633901066873595346096298659855593548453567339620090693131651503331290224972643248428821639173717842799629028021042107274754690384076727696817482166801994662813450060403484823724401, + 602142912206973711887315561170904756952640210127850551940789493157746056377379826763616362402020237696865327014080068950216304555089934702434312544962569240809623049749328049167909681727688996310217153579438228870667754675519567228647196647974878833696933930210200163871410118653330480389842312109713005945868498493675588732493636994540139751795974102904699839249296328754059282438334102079705540750359815808000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 + ] + } + }, + "fivefold_followup": { + "claim_boundary": "No 78-step global error certificate is claimed or supplied.", + "conditional_exact_ratio": [ + 11791, + 2341 + ], + "conditional_group_exponentials": 2341, + "conditional_ratio_decimal": "5.036736437419906", + "conditional_step_count": 78, + "d5_group_count": 123106, + "d5_site_density_upper": [ + 44948270001027856175670154896253, + 4000000000000000000000000000000 + ], + "d5_site_density_upper_decimal": "1.123706750025697e+1", + "d5_term_count": 605832, + "status": "not_certified", + "unresolved_proof_gates": [ + "translation-coupled D4 norm below the r=78 budget", + "explicit D8 and delayed high-degree tail certificate", + "complete exact global error ledger at r=78" + ] + }, + "improvement": { + "decimal": "4.050498110614909", + "exact_ratio": [ + 11791, + 2911 + ], + "multiple_of_required_twofold_target": "2.025249055307454", + "resource_fraction_removed": [ + 8880, + 11791 + ], + "resource_percent_removed": "75.31167839877873%" + }, + "method": { + "innovations": [ + "full free-associative-word logarithm before norm inequalities", + "Dynkin-Specht-Wever projection to degree-5 and degree-7 Lie elements", + "translation-canonical Pauli combination before taking norms", + "exact pairwise-anticommuting grouping for the leading D4 defect", + "exact single-bond Heisenberg commutator growth constant 1", + "finite-step right-generator ledger with a rational geometric tail", + "independent exact-rational and symplectic verification" + ], + "proof_method": "local_log_E5_grouped_D4_plus_E7_majorant_plus_exact_generator_tail" + }, + "published_baseline": { + "bond_propagators": 848952, + "cnot_upper": 2546856, + "group_exponentials": 11791, + "site_density_upper": [ + 13198073356219512631693646217169939173745709526165852646345503355718759045842787628228078578619, + 80000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 + ], + "site_density_upper_decimal": "1.649759169527440e+2", + "source_theorem": "Childs_Su_Tran_Wiebe_Zhu_high_order_commutator_bound", + "steps": 393, + "theorem_center": 20 + }, + "schema_version": 1, + "sources": { + "anchored_pr_head": "c550a6b0915ccf7a1db410ee765ff65d99f96b6f", + "d4_sidecar": { + "path": "certificates/issue128-d4-groups.json", + "sha256": "a397414bb0229fb1ebdb38798aa781fb89dbb9d5cdbed94c7cd2e9120da62718" + }, + "d5_sidecar": { + "path": "certificates/issue128-d5-groups.json.gz", + "sha256": "c5e8968a93b4497b41fe42c0e364324388272e183e1fcd20a536bc988f5361dd" + }, + "main_certificate": { + "path": "certificates/issue128-certificate.json", + "sha256": "0a09623ce3b292a3637065c870fb3153bbdcddce30aef968565c4db3ddfc7201" + }, + "source_commit_for_main_certificate": "d7381a630b1e42194fb6635ffb0864ca326db0e2", + "source_commit_timestamp": "2026-07-29T21:54:48+08:00" + }, + "verification": { + "d4_cell_norm_upper": [ + 6472926505087887788180917647763, + 1000000000000000000000000000000 + ], + "d4_cell_norm_upper_decimal": "6.472926505087888e+0", + "d4_group_count": 7576, + "d4_max_group_size": 10, + "d4_term_count": 75324, + "small_crosscheck": { + "bound_dominates_empirical_error": true, + "bound_meets_tolerance": true, + "certified_upper_bound": 2.7666666666666666e-08, + "empirical_operator_norm_error": 9.128277711816321e-10, + "length": 2, + "model_note": "degenerate 2x2 periodic algebra sanity check for schema-v3 r=97", + "n_sites": 4, + "requested_tolerance": 1e-06, + "steps": 97 + } + } +} diff --git a/tracks/qcs/solutions/WangTheoPhys/issue128/artifacts/issue128-summary.txt b/tracks/qcs/solutions/WangTheoPhys/issue128/artifacts/issue128-summary.txt new file mode 100644 index 000000000..0002e8339 --- /dev/null +++ b/tracks/qcs/solutions/WangTheoPhys/issue128/artifacts/issue128-summary.txt @@ -0,0 +1,28 @@ +ISSUE 128 FINAL RESULT SUMMARY +================================ + +BENCHMARK: periodic_square_spin_half_isotropic_heisenberg +LATTICE: L=12, N=144 +NORMALIZATION: (XX+YY+ZZ)/4 +TIME: T=1; OPERATOR-NORM TOLERANCE: 1e-6 + +PUBLISHED BASELINE: 393 steps, 11791 groups, 848952 bond propagators, 2546856 CNOT upper +CERTIFIED RESULT: 4.050498110614909x + 97 steps, 2911 groups, 209592 bond propagators, 628776 CNOT upper + exact ratio: 11791/2911 + resource reduction: 75.31167839877873% + 97-step error upper (outward decimal): 9.958938494314325e-7 + 96-step error upper (outward decimal): 1.050565873970784e-6 + integer boundary: 97 accepted; 96 rejected + +EXACT D4: 75324 terms, 7576 groups, maximum size 10, cell bound 6.472926505087888e+0 + +FIVEFOLD STATUS: NOT CERTIFIED + conditional arithmetic only: r=78, groups=2341, ratio=5.036736437419906x + verified D5 evidence: 605832 terms, 123106 groups, site bound 1.123706750025697e+1 + No 78-step global error certificate is claimed or supplied. + +SOURCE BINDINGS + main certificate SHA-256: 0a09623ce3b292a3637065c870fb3153bbdcddce30aef968565c4db3ddfc7201 + D4 sidecar SHA-256: a397414bb0229fb1ebdb38798aa781fb89dbb9d5cdbed94c7cd2e9120da62718 + D5 sidecar SHA-256: c5e8968a93b4497b41fe42c0e364324388272e183e1fcd20a536bc988f5361dd diff --git a/tracks/qcs/solutions/WangTheoPhys/issue128/artifacts/verification-transcript.txt b/tracks/qcs/solutions/WangTheoPhys/issue128/artifacts/verification-transcript.txt new file mode 100644 index 000000000..680bc07b6 --- /dev/null +++ b/tracks/qcs/solutions/WangTheoPhys/issue128/artifacts/verification-transcript.txt @@ -0,0 +1,43 @@ +ISSUE 128 VERIFICATION TRANSCRIPT +================================= +source_commit_for_main_certificate: d7381a630b1e42194fb6635ffb0864ca326db0e2 +source_commit_timestamp: 2026-07-29T21:54:48+08:00 +python: 3.12.11 +platform: macOS-15.6-arm64-arm-64bit + +$ PYTHONPATH=src python scripts/verify.py certificates/issue128-certificate.json +exit_code: 0 +--- stdout --- +{ + "valid": true, + "verification_level": "fast", + "deep_proof_regenerated": false, + "published_steps": 393, + "candidate_steps": 97, + "published_group_exponentials": 11791, + "candidate_group_exponentials": 2911, + "candidate_error_upper": "599670422745656108926597685630153943951201095508880641428475695866175124754901808839128699565476683604156496265888897758053359535162204152523497215954848848755961894315360438774573129074241941927331478665448272993350597926513476349231331606355445437094033812134866153343154999415560759372473990440779968348417384816501377794329390792937346394068204978527979293780943439329901747152018428695684650908057947950752275565153549211215094936727050696808115471285220887575343320538864415440435484823724401/602142912206973711887315561170904756952640210127850551940789493157746056377379826763616362402020237696865327014080068950216304555089934702434312544962569240809623049749328049167909681727688996310217153579438228870667754675519567228647196647974878833696933930210200163871410118653330480389842312109713005945868498493675588732493636994540139751795974102904699839249296328754059282438334102079705540750359815808000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000", + "previous_step_error_upper": "18628098815762331308955745375744031451312712544003951915143323587191765140913687282590556923012229284286727492973637946908091979520580506941522631107558919299568456616026386835101726158947303895942669287179491573040382549259481731100862869066053794714470474282725565473839414726457780840434975235932140103612890697147056306440018341055946734378261970229499903599140373782008230578526026304631306088724623197505724600262101991677390833449941204813309538996425837404480621025826120161090811331145602190211/17731490501737330132971023158265461256557754955863122336739410011121346612799210695034681326844820557504793235783494060750359613343514225496752334858555018334128350507379874818501208678928093800230834997231437643227318036143974224856394241661177886302136098849703389670867925933947203616654322290404728008558748552082205606402627773932681885052986146827660046233143704985663445575306347781684791090345692853436416000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000", + "d4_cell_norm_upper": "6472926505087887788180917647763/1000000000000000000000000000000", + "d4_term_count": 75324, + "d4_group_count": 7576, + "d4_max_group_size": 10, + "exact_improvement_ratio": "11791/2911", + "improvement": 4.050498110614909, + "global_twofold_target_met": true, + "global_fourfold_target_met": true +} +--- stderr --- +(empty) + +$ PYTHONPATH=src python scripts/build_d5_certificate.py --verify-only +exit_code: 0 +--- stdout --- +term_count=605832 +group_count=123106 +site_density=44948270001027856175670154896253/4000000000000000000000000000000 +site_density_decimal=11.23706750025696 +--- stderr --- +(empty) + +OVERALL STATUS: PASS diff --git a/tracks/qcs/solutions/WangTheoPhys/issue128/certificates/issue128-certificate.json b/tracks/qcs/solutions/WangTheoPhys/issue128/certificates/issue128-certificate.json new file mode 100644 index 000000000..c988161f9 --- /dev/null +++ b/tracks/qcs/solutions/WangTheoPhys/issue128/certificates/issue128-certificate.json @@ -0,0 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diff --git a/tracks/qcs/solutions/WangTheoPhys/issue128/certificates/issue128-d5-groups.json.gz b/tracks/qcs/solutions/WangTheoPhys/issue128/certificates/issue128-d5-groups.json.gz new file mode 100644 index 000000000..2ffd6435e Binary files /dev/null and b/tracks/qcs/solutions/WangTheoPhys/issue128/certificates/issue128-d5-groups.json.gz differ diff --git a/tracks/qcs/solutions/WangTheoPhys/issue128/certificates/issue128-degree3-rank.json b/tracks/qcs/solutions/WangTheoPhys/issue128/certificates/issue128-degree3-rank.json new file mode 100644 index 000000000..da76e6d93 --- /dev/null +++ b/tracks/qcs/solutions/WangTheoPhys/issue128/certificates/issue128-degree3-rank.json @@ -0,0 +1,10 @@ +{ + "model": "periodic_4x4_square_spin_half_heisenberg", + "decomposition": "four_disjoint_bond_matchings", + "operators": "[H_a,[H_b,H_c]] for a,b,c in {0,1,2,3}", + "columns": 64, + "pauli_rows": 4128, + "exact_rational_rank": 20, + "free_lie_degree_three_dimension_for_four_generators": 20, + "representation_specific_rank_reduction": false +} diff --git a/tracks/qcs/solutions/WangTheoPhys/issue128/certificates/issue128-small-crosscheck.json b/tracks/qcs/solutions/WangTheoPhys/issue128/certificates/issue128-small-crosscheck.json new file mode 100644 index 000000000..4405cc81b --- /dev/null +++ b/tracks/qcs/solutions/WangTheoPhys/issue128/certificates/issue128-small-crosscheck.json @@ -0,0 +1,11 @@ +{ + "length": 2, + "n_sites": 4, + "model_note": "degenerate 2x2 periodic algebra sanity check for schema-v3 r=97", + "steps": 97, + "empirical_operator_norm_error": 9.128277711816321e-10, + "certified_upper_bound": 2.7666666666666666e-08, + "requested_tolerance": 1e-06, + "bound_dominates_empirical_error": true, + "bound_meets_tolerance": true +} diff --git a/tracks/qcs/solutions/WangTheoPhys/issue128/certificates/issue128-three-cluster-certificate.json b/tracks/qcs/solutions/WangTheoPhys/issue128/certificates/issue128-three-cluster-certificate.json new file mode 100644 index 000000000..33f6b3e34 --- /dev/null +++ b/tracks/qcs/solutions/WangTheoPhys/issue128/certificates/issue128-three-cluster-certificate.json @@ -0,0 +1,42 @@ +{ + "schema_version": 1, + "benchmark": { + "model": "periodic_square_spin_half_isotropic_heisenberg", + "normalization": "(XX+YY+ZZ)/4", + "length": 12, + "time": [1, 1], + "tolerance": [1, 1000], + "primary_certified_metric": "local_propagator_count", + "secondary_metric": "generic_cnot_upper" + }, + "proof": { + "reference_length": 6, + "scaling": "translation-invariant density times N", + "baseline_decomposition": "four_disjoint_bond_matchings", + "candidate_decomposition": "three_disjoint_L_path_cluster_colors", + "baseline_density": [21, 8], + "candidate_density": [13, 6], + "formula": "second_order_symmetric", + "single_step_bound": "C2*t^3", + "multi_step_bound": "C2*T^3/r^2" + }, + "cost_model": { + "baseline_local_support": "two-site Heisenberg bond", + "candidate_local_support": "three-site L path with two Heisenberg bonds", + "baseline_cnots_per_local_propagator": 3, + "candidate_cnots_per_local_propagator": 9, + "candidate_cnot_value_is_a_generic_encoder_plus_SU4_upper_bound": true + }, + "claimed_resources": { + "baseline_steps": 615, + "candidate_steps": 559, + "baseline_local_propagators": 265752, + "candidate_local_propagators": 107376, + "baseline_cnot_upper": 797256, + "candidate_cnot_upper": 966384 + }, + "claims": { + "local_propagator_improvement_exceeds_two": true, + "compiled_cnot_improvement_exceeds_two": false + } +} diff --git a/tracks/qcs/solutions/WangTheoPhys/issue128/certificates/issue128-verification.txt b/tracks/qcs/solutions/WangTheoPhys/issue128/certificates/issue128-verification.txt new file mode 100644 index 000000000..a88b1069b --- /dev/null +++ b/tracks/qcs/solutions/WangTheoPhys/issue128/certificates/issue128-verification.txt @@ -0,0 +1,91 @@ +Issue #128 deep verification transcript +======================================= + +Status +------ +valid: true +verification_level: deep +deep_proof_regenerated: true +test_suite: 70 passed, 9 slow tests deselected (Python 3.12) + +Benchmark +--------- +Hamiltonian: periodic square-lattice spin-1/2 isotropic Heisenberg +normalization: h_ij = (XX+YY+ZZ)/4 +L: 12 +N: 144 +T: 1 +operator-norm tolerance: 1e-6 +cost: group exponentials, bond propagators, and 3-CNOT-per-bond upper bound + +Published rigorous baseline +--------------------------- +formula: five-copy fourth-order Suzuki formula +published theorem: Childs-Su-Tran-Wiebe-Zhu high-order commutator bound +norm instantiation: expand each partial Hamiltonian sum, then local Pauli-l1 +all 31 theorem centers scanned: center 20 is minimal +certified site-density upper bound: 164.97591695274392 +steps: 393 +group exponentials: 11791 +bond propagators: 848952 +CNOT upper bound: 2546856 + +Candidate certificate +--------------------- +formula: the same five-copy fourth-order Suzuki formula +method: local log E5 + grouped D4 + E7 majorant + exact right-generator tail +certified E5 site Pauli-l1 upper bound: 1.0312915120959002 +certified E7 site majorant: 847.4072568943714 +exact bond Pauli-l1 commutator growth: 1 +local proof: all 16 phase-free two-qubit Pauli cases enumerated exactly +canonical D4 Pauli terms: 75324 +pairwise-anticommuting groups: 7576 +maximum group size: 10 +grouped D4 cell-norm upper bound: 6.472926505087888 +sidecar SHA-256: a397414bb0229fb1ebdb38798aa781fb89dbb9d5cdbed94c7cd2e9120da62718 + +At r=97: +degree-4 contribution: 5.264367936822258e-7 +degree-5 contribution: 1.3834875367446118e-7 +degree-6 contribution: 1.579057000594835e-7 +degree-7 contribution: 1.8946570806914694e-8 +degree >= 8 tail: 1.5425603120834724e-7 +global error upper bound: 9.958938494314325e-7 + +At r=96: +global error upper bound: 1.050565873970784e-6 + +Therefore r=97 is the smallest step count accepted by this certificate. + +Candidate resources +------------------- +steps: 97 +group exponentials: 2911 +bond propagators: 209592 +CNOT upper bound: 628776 + +Certified comparison +-------------------- +exact group/CNOT improvement ratio: 11791/2911 +decimal improvement: 4.050498110614909 +global twofold target met: true +global fourfold target met: true + +Additional published recursive-Suzuki audit +------------------------------------------- +The same generic locality tail theorem gives: + +order steps stages/step merged group exponentials +2 44946 7 269677 +4 1212 31 36361 +6 554 151 83101 +8 503 751 377251 + +These are conservative generic-order bounds, not replacements for the +sharper Childs fourth-order baseline above. They establish that simply +raising the recursive Suzuki order does not create a closer rigorous +competitor for this benchmark. + +Verifier command +---------------- +PYTHONPATH=src python scripts/verify.py certificates/issue128-certificate.json --deep diff --git a/tracks/qcs/solutions/WangTheoPhys/issue128/docs/followups/2026-07-29-5x-10x-audit.md b/tracks/qcs/solutions/WangTheoPhys/issue128/docs/followups/2026-07-29-5x-10x-audit.md new file mode 100644 index 000000000..670ef060d --- /dev/null +++ b/tracks/qcs/solutions/WangTheoPhys/issue128/docs/followups/2026-07-29-5x-10x-audit.md @@ -0,0 +1,118 @@ +# Issue #128: 5× and 10× certification follow-up audit + +## Starting point + +The submitted certificate fixes the benchmark and circuit: + +```text +L = 12, N = 144, T = 1, epsilon = 1e-6 +five-copy fourth-order Suzuki formula +merged group exponentials = 30 r + 1 +published rigorous baseline = 11791 groups +``` + +The grouped-D4 certificate accepts `r=97` and rejects `r=96`, giving + +```text +11791 / 2911 = 4.050498110614909... +``` + +The following audit keeps the same exact-rational error ledger and asks what +would be required to reach the next resource thresholds. + +## Integer resource thresholds + +Fivefold improvement requires + +```text +30 r + 1 <= floor(11791 / 5), +``` + +so `r <= 78`. Tenfold improvement similarly requires `r <= 39`. + +## Error ledger at the thresholds + +The degree-four through degree-seven terms are rescaled exactly from their +certified `r=97` values. The geometric right-generator tail is reevaluated at +each step count. + +| Contribution | `r=78` (5× threshold) | `r=39` (10× threshold) | +|---|---:|---:| +| Degree 4 | `1.2590841639313686e-6` | `2.0145346622901898e-5` | +| Degree 5 | `4.1149153826738221e-7` | `1.3167729224556231e-5` | +| Degree 6 | `5.8406421306735928e-7` | `3.7380109636310994e-5` | +| Degree 7 | `8.7150631476060649e-8` | `1.1155280828935763e-5` | +| Degree 8 and above | `9.0076387927122002e-7` | `2.580807568220183e-4` | +| **Total** | **`3.2425544260133909e-6`** | **`3.399292231347232e-4`** | +| Total with degree 4 set to zero | `1.9834702620820221e-6` | `3.1978387651182126e-4` | + +## Consequences + +### Fivefold improvement + +Further tightening only the grouped D4 norm cannot reach fivefold +improvement: even an impossible zero D4 contribution leaves +`1.9834702620820221e-6`, above the target. + +The main remaining bottleneck at `r=78` is the degree-eight-and-above +geometric tail (`9.0076e-7`). It nearly exhausts the complete `1e-6` budget +before degrees five through seven are counted. A credible 5× attempt therefore +needs both: + +1. exact canonical coefficient maps and anticommuting/cluster certificates + for D5, D6, and D7; and +2. explicit certification of D8-D10 (or farther), postponing the geometric + tail to a sufficiently high degree. + +This is still plausibly a proof improvement with the same circuit. The +go/no-go test is direct: rebuild the exact ledger at `r=78`; continue only if +the certified sum is at most `1e-6`. + +### Tenfold improvement + +At `r=39`, eliminating D4 entirely still leaves +`3.1978387651182126e-4`, requiring a further factor of about `319.8`. +The tail alone is `2.5808e-4`. Norm tightening within the present +fourth-order ledger is therefore not a realistic route to 10×. + +Tenfold improvement requires changing the low-degree error structure: + +- a local corrector/processor that cancels the leading logarithmic error; +- a model-specific higher-order composition; or +- a shorter sixth-order kernel with a rigorously smaller finite-step tail. + +For the present fourth-order kernel, + +```text +log S(t) = t H + t^5 B + O(t^7). +``` + +A processor `P(t)=exp(t^4 Q)` changes the leading error by a term proportional +to `[Q,H]`. The first exact obstruction test is therefore a rational +translation-orbit linear-algebra problem: + +```text +Does B lie in image(ad_H) for a finite-range SU(2)-invariant/chirality basis Q? +``` + +If yes, certify the processed fifth- and higher-degree defect and include the +two endpoint processor costs. If not, record the exact residual rank and move +to model-specific stage optimization or a sixth-order kernel. + +## Recommended implementation order + +1. Replace the memory-heavy general local-series expansion with a streamed + integer/symplectic recurrence for canonical D5-D10 coefficient maps. +2. Reuse the existing exact anticommuting sidecar format for each degree. +3. Reevaluate `r=78`; this is the 5× decision point. +4. In parallel, build the exact `image(ad_H)` rank test for chirality and + short-range SU(2)-invariant processor bases. +5. Only if the leading error is exactly cancellable, implement the processed + finite-step certificate and compilation ledger. +6. Otherwise search short symmetric sixth-order formulas, screening candidates + with local leading-error norms before interval certification. + +The 4.050× result is complete without these follow-ups. This audit separates +the next proof-tightening target (5×) from the order-lift target (10×) and +prevents spending further effort on D4 optimization that cannot cross either +integer boundary by itself. diff --git a/tracks/qcs/solutions/WangTheoPhys/issue128/docs/manuscript/CITATION_AUDIT.md b/tracks/qcs/solutions/WangTheoPhys/issue128/docs/manuscript/CITATION_AUDIT.md new file mode 100644 index 000000000..cbeb6d903 --- /dev/null +++ b/tracks/qcs/solutions/WangTheoPhys/issue128/docs/manuscript/CITATION_AUDIT.md @@ -0,0 +1,23 @@ +# Citation audit + +Audit date: 2026-07-31 (Asia/Shanghai) + +The compiled manuscript contains 23 references. All 23 were checked by stable +identifier and title before finalization. + +| Validation route | Count | Result | +|---|---:|---| +| DOI queried through OpenAlex | 13 | 13 exact title/DOI matches | +| Modern arXiv identifier queried through the arXiv API | 8 | 8 exact title/ID matches | +| Legacy arXiv identifier (`quant-ph/9705052`) | 1 | Exact title, author, and year match | +| Oxford Academic official catalog | 1 | Exact book title, author, year, publisher, ISBN, and DOI match | + +The book not returned by the OpenAlex title search was Christophe Reutenauer, +*Free Lie Algebras*. It was verified against the +[Oxford Academic catalog](https://academic.oup.com/book/53911): published May +6, 1993 by Oxford University Press, print ISBN `9780198536796`, DOI +`10.1093/oso/9780198536796.001.0001`. + +No citation was classified as not found, mismatched, or invalid. The PDF's +reference list is generated from `references.bib`; inline claims about the +literature remain deliberately scoped by the comparison taxonomy in the paper. diff --git a/tracks/qcs/solutions/WangTheoPhys/issue128/docs/manuscript/Makefile b/tracks/qcs/solutions/WangTheoPhys/issue128/docs/manuscript/Makefile new file mode 100644 index 000000000..e115ba030 --- /dev/null +++ b/tracks/qcs/solutions/WangTheoPhys/issue128/docs/manuscript/Makefile @@ -0,0 +1,16 @@ +PYTHON ?= python3 +LATEXMK ?= latexmk +JOBNAME := issue128-proof-carrying-trotter-paper +OUTDIR := output/pdf + +.PHONY: check figures pdf + +check: + $(PYTHON) scripts/validate_claims.py + +figures: + $(PYTHON) scripts/generate_figures.py + +pdf: check figures + $(LATEXMK) -pdf -interaction=nonstopmode -halt-on-error \ + -jobname=$(JOBNAME) -outdir=$(OUTDIR) main.tex diff --git a/tracks/qcs/solutions/WangTheoPhys/issue128/docs/manuscript/README.md b/tracks/qcs/solutions/WangTheoPhys/issue128/docs/manuscript/README.md new file mode 100644 index 000000000..ebd41677c --- /dev/null +++ b/tracks/qcs/solutions/WangTheoPhys/issue128/docs/manuscript/README.md @@ -0,0 +1,54 @@ +# Long-form Issue 128 manuscript + +This directory contains the full English research-paper treatment of the +machine-certified Issue 128 result. The authoritative exact numbers remain in +`../../certificates/issue128-certificate.json` and its hashed sidecars; the PDF +is a human-readable derived artifact. + +## Build + +From this directory: + +```bash +make pdf +``` + +The final file is: + +```text +output/pdf/issue128-proof-carrying-trotter-paper.pdf +``` + +The build first validates headline claims against the frozen JSON, regenerates +the four vector PDF figures, and then runs `latexmk` with BibTeX until all +cross-references settle. + +Equivalent commands are: + +```bash +python3 scripts/validate_claims.py +python3 scripts/generate_figures.py +latexmk -pdf -interaction=nonstopmode -halt-on-error \ + -jobname=issue128-proof-carrying-trotter-paper \ + -outdir=output/pdf main.tex +``` + +## Verification levels + +From the Issue 128 root: + +```bash +pytest -q +PYTHONPATH=src python3 scripts/verify.py \ + certificates/issue128-certificate.json +``` + +The expensive independent replay is optional: + +```bash +PYTHONPATH=src python3 scripts/verify.py \ + certificates/issue128-certificate.json --deep +``` + +See `CITATION_AUDIT.md` for the reference-validation result. Generated LaTeX +auxiliary files are ignored; the named final PDF is retained. diff --git a/tracks/qcs/solutions/WangTheoPhys/issue128/docs/manuscript/figures/d4_norm.pdf b/tracks/qcs/solutions/WangTheoPhys/issue128/docs/manuscript/figures/d4_norm.pdf new file mode 100644 index 000000000..3290334f7 Binary files /dev/null and b/tracks/qcs/solutions/WangTheoPhys/issue128/docs/manuscript/figures/d4_norm.pdf differ diff --git a/tracks/qcs/solutions/WangTheoPhys/issue128/docs/manuscript/figures/error_ledger.pdf b/tracks/qcs/solutions/WangTheoPhys/issue128/docs/manuscript/figures/error_ledger.pdf new file mode 100644 index 000000000..0c504e8d7 Binary files /dev/null and b/tracks/qcs/solutions/WangTheoPhys/issue128/docs/manuscript/figures/error_ledger.pdf differ diff --git a/tracks/qcs/solutions/WangTheoPhys/issue128/docs/manuscript/figures/fivefold_gap.pdf b/tracks/qcs/solutions/WangTheoPhys/issue128/docs/manuscript/figures/fivefold_gap.pdf new file mode 100644 index 000000000..92062f403 Binary files /dev/null and b/tracks/qcs/solutions/WangTheoPhys/issue128/docs/manuscript/figures/fivefold_gap.pdf differ diff --git a/tracks/qcs/solutions/WangTheoPhys/issue128/docs/manuscript/figures/resources.pdf b/tracks/qcs/solutions/WangTheoPhys/issue128/docs/manuscript/figures/resources.pdf new file mode 100644 index 000000000..6553ea15f Binary files /dev/null and b/tracks/qcs/solutions/WangTheoPhys/issue128/docs/manuscript/figures/resources.pdf differ diff --git a/tracks/qcs/solutions/WangTheoPhys/issue128/docs/manuscript/main.tex b/tracks/qcs/solutions/WangTheoPhys/issue128/docs/manuscript/main.tex new file mode 100644 index 000000000..fd6b73f8f --- /dev/null +++ b/tracks/qcs/solutions/WangTheoPhys/issue128/docs/manuscript/main.tex @@ -0,0 +1,107 @@ +\documentclass[11pt]{article} + +\usepackage[margin=0.82in]{geometry} +\usepackage[T1]{fontenc} +\usepackage[utf8]{inputenc} +\usepackage{lmodern} +\usepackage{microtype} +\usepackage{amsmath,amssymb,amsthm,mathtools} +\usepackage{booktabs,tabularx,array,longtable,multirow} +\usepackage{xcolor} +\usepackage{graphicx} +\usepackage{enumitem} +\usepackage{algorithm} +\usepackage{algpseudocode} +\usepackage{siunitx} +\usepackage{tikz} +\usetikzlibrary{arrows.meta,positioning,fit,backgrounds} +\usepackage[numbers,sort&compress]{natbib} +\usepackage{hyperref} +\usepackage[nameinlink,noabbrev]{cleveref} +\usepackage{fancyhdr} +\makeatletter +\renewcommand{\theALG@line}{\thealgorithm.\arabic{ALG@line}} +\providecommand*{\theHALG@line}{} +\renewcommand*{\theHALG@line}{\thealgorithm.\arabic{ALG@line}} +\makeatother + +\definecolor{certblue}{HTML}{2455A4} +\definecolor{certgreen}{HTML}{2A7F62} +\definecolor{certorange}{HTML}{D97706} +\definecolor{certred}{HTML}{B42318} +\hypersetup{ + colorlinks=true, + linkcolor=certblue, + citecolor=certgreen, + urlcolor=certblue, + pdftitle={Proof-Carrying, Model-Aware Compilation of Rigorous Trotter Resource Bounds}, + pdfauthor={Junkai Wang} +} + +\pagestyle{fancy} +\fancyhf{} +\fancyhead[L]{Proof-Carrying Trotter Resource Bounds} +\fancyhead[R]{J. Wang} +\fancyfoot[C]{\thepage} +\setlength{\headheight}{14pt} +\setlength{\parindent}{1.25em} +\setlength{\parskip}{0.22em} +\setlength{\emergencystretch}{3em} +\setlist{itemsep=0.25em,topsep=0.35em} + +\newtheorem{theorem}{Theorem}[section] +\newtheorem{lemma}[theorem]{Lemma} +\newtheorem{proposition}[theorem]{Proposition} +\newtheorem{corollary}[theorem]{Corollary} +\theoremstyle{definition} +\newtheorem{definition}[theorem]{Definition} +\newtheorem{remark}[theorem]{Remark} + +\newcommand{\opnorm}[1]{\left\lVert #1\right\rVert} +\newcommand{\paulinorm}[1]{\left\lVert #1\right\rVert_{1,\mathrm{P}}} +\newcommand{\comm}[2]{\left[#1,#2\right]} +\newcommand{\ad}{\operatorname{ad}} +\newcommand{\eps}{\varepsilon} +\newcommand{\cert}{\textsc{Cert}} +\newcommand{\code}[1]{\texttt{#1}} +\newcommand{\Rge}{R_{\ge 8}} + +\title{\textbf{Proof-Carrying, Model-Aware Compilation of\\ +Rigorous Trotter Resource Bounds}\\[0.4em] +\large A Machine-Certified $4.050498\times$ Reduction for a\\ +Two-Dimensional Heisenberg Benchmark} +\author{Junkai Wang\\ +\small WangTheoPhys / Wander Team\\ +\small Quantum Circuit Simulation Track\\ +\small \href{mailto:WangTheoPhys@outlook.com}{WangTheoPhys@outlook.com}} +\date{July 31, 2026} + +\begin{document} +\maketitle +\input{sections/abstract} +\tableofcontents +\clearpage + +\input{sections/introduction} +\input{sections/problem} +\input{sections/baseline} +\input{sections/compiler} +\input{sections/free_lie} +\input{sections/pauli} +\input{sections/norm_certificate} +\input{sections/finite_step} +\input{sections/verification} +\input{sections/results} +\input{sections/related_work} +\input{sections/limitations} +\input{sections/reproducibility} +\input{sections/conclusion} + +\appendix +\input{sections/appendices} + +{\small +\bibliographystyle{unsrtnat} +\bibliography{references} +} +\end{document} diff --git a/tracks/qcs/solutions/WangTheoPhys/issue128/docs/manuscript/output/pdf/.gitignore b/tracks/qcs/solutions/WangTheoPhys/issue128/docs/manuscript/output/pdf/.gitignore new file mode 100644 index 000000000..1f04081b4 --- /dev/null +++ b/tracks/qcs/solutions/WangTheoPhys/issue128/docs/manuscript/output/pdf/.gitignore @@ -0,0 +1,3 @@ +* +!.gitignore +!issue128-proof-carrying-trotter-paper.pdf diff --git a/tracks/qcs/solutions/WangTheoPhys/issue128/docs/manuscript/output/pdf/issue128-proof-carrying-trotter-paper.pdf b/tracks/qcs/solutions/WangTheoPhys/issue128/docs/manuscript/output/pdf/issue128-proof-carrying-trotter-paper.pdf new file mode 100644 index 000000000..bf74213d6 Binary files /dev/null and b/tracks/qcs/solutions/WangTheoPhys/issue128/docs/manuscript/output/pdf/issue128-proof-carrying-trotter-paper.pdf differ diff --git a/tracks/qcs/solutions/WangTheoPhys/issue128/docs/manuscript/references.bib b/tracks/qcs/solutions/WangTheoPhys/issue128/docs/manuscript/references.bib new file mode 100644 index 000000000..0897b264d --- /dev/null +++ b/tracks/qcs/solutions/WangTheoPhys/issue128/docs/manuscript/references.bib @@ -0,0 +1,135 @@ +@article{childs2021trotter, + title={Theory of Trotter Error with Commutator Scaling}, + author={Childs, Andrew M. and Su, Yuan and Tran, Minh C. and Wiebe, Nathan and Zhu, Shuchen}, + journal={Physical Review X}, + volume={11}, number={1}, pages={011020}, year={2021}, + doi={10.1103/PhysRevX.11.011020}, eprint={1912.08854}, archivePrefix={arXiv} +} +@article{childs2019lattice, + title={Nearly Optimal Lattice Simulation by Product Formulas}, + author={Childs, Andrew M. and Su, Yuan}, + journal={Physical Review Letters}, volume={123}, pages={050503}, year={2019}, + doi={10.1103/PhysRevLett.123.050503} +} +@article{schubert2023hubbard, + title={Trotter Error with Commutator Scaling for the {Fermi--Hubbard} Model}, + author={Schubert, Ansgar and Mendl, Christian B.}, + journal={arXiv preprint arXiv:2306.10603}, year={2023}, + eprint={2306.10603}, archivePrefix={arXiv}, primaryClass={quant-ph} +} +@article{sahinoglu2021lowenergy, + title={Hamiltonian Simulation in the Low-Energy Subspace}, + author={\c{S}ahino\u{g}lu, Burak and Somma, Rolando D.}, + journal={npj Quantum Information}, volume={7}, pages={119}, year={2021}, + doi={10.1038/s41534-021-00451-w} +} +@article{chen2024average, + title={Average-Case Speedup for Product Formulas}, + author={Chen, Chi-Fang and Brand\~ao, Fernando G. S. L.}, + journal={Communications in Mathematical Physics}, volume={405}, pages={32}, year={2024}, + doi={10.1007/s00220-023-04912-5} +} +@article{zhuk2024dynamic, + title={Trotter Error Bounds and Dynamic Multi-Product Formulas for Hamiltonian Simulation}, + author={Zhuk, Sergiy and Robertson, Niall F. and Bravyi, Sergey}, + journal={Physical Review Research}, volume={6}, pages={033309}, year={2024}, + doi={10.1103/PhysRevResearch.6.033309} +} +@article{aftab2024mpf, + title={Multi-Product Hamiltonian Simulation with Explicit Commutator Scaling}, + author={Aftab, Junaid and An, Dong and Trivisa, Konstantina}, + journal={arXiv preprint arXiv:2403.08922}, year={2024}, + eprint={2403.08922}, archivePrefix={arXiv}, primaryClass={quant-ph} +} +@article{rendon2024chebyshev, + title={Improved Accuracy for Trotter Simulations Using Chebyshev Interpolation}, + author={Rendon, Gumaro and Watkins, Jacob and Wiebe, Nathan}, + journal={Quantum}, volume={8}, pages={1266}, year={2024}, + doi={10.22331/q-2024-02-26-1266} +} +@article{cho2024doubling, + title={Doubling the Order of Approximation via the Randomized Product Formula}, + author={Cho, Chien-Hung and Berry, Dominic W. and Hsieh, Min-Hsiu}, + journal={Physical Review A}, volume={109}, pages={062431}, year={2024}, + doi={10.1103/PhysRevA.109.062431} +} +@article{burgarth2024strong, + title={Strong Error Bounds for Trotter and Strang Splittings and Their Implications for Quantum Chemistry}, + author={Burgarth, Daniel and Facchi, Paolo and Hahn, Alexander and Johnsson, Mattias and Yuasa, Kazuya}, + journal={Physical Review Research}, volume={6}, pages={043155}, year={2024}, + doi={10.1103/PhysRevResearch.6.043155} +} +@article{hahn2024lower, + title={Lower Bounds for the Trotter Error}, + author={Hahn, Alexander and Hartung, Paul and Burgarth, Daniel and Facchi, Paolo and Yuasa, Kazuya}, + journal={arXiv preprint arXiv:2410.03059}, year={2024}, + eprint={2410.03059}, archivePrefix={arXiv}, primaryClass={quant-ph} +} +@article{bosse2025thrift, + title={Efficient and Practical Hamiltonian Simulation from Time-Dependent Product Formulas}, + author={Bosse, Jan Lukas and Childs, Andrew M. and Derby, Charles and Gambetta, Filippo Maria and Montanaro, Ashley and Santos, Raul A.}, + journal={Nature Communications}, volume={16}, pages={2673}, year={2025}, + doi={10.1038/s41467-025-57580-5} +} +@article{zeng2025lcu, + title={Simple and High-Precision Hamiltonian Simulation by Compensating Trotter Error with Linear Combination of Unitary Operations}, + author={Zeng, Pei and Sun, Jinzhao and Jiang, Liang and Zhao, Qi}, + journal={PRX Quantum}, volume={6}, pages={010359}, year={2025}, + doi={10.1103/PRXQuantum.6.010359} +} +@article{watson2025mitigation, + title={Exponentially Reduced Circuit Depths Using Trotter Error Mitigation}, + author={Watson, James D. and Watkins, Jacob}, + journal={PRX Quantum}, volume={6}, pages={030325}, year={2025}, + doi={10.1103/kw39-yxq5} +} +@article{mizuta2026mpf, + title={On the Commutator Scaling in Hamiltonian Simulation with Multi-Product Formulas}, + author={Mizuta, Kaoru}, + journal={Quantum}, volume={10}, pages={1974}, year={2026}, + doi={10.22331/q-2026-01-19-1974} +} +@article{mizuta2026timedependent, + title={Explicit Error Bounds with Commutator Scaling for Time-Dependent Product and Multi-Product Formulas}, + author={Mizuta, Kaoru and Ikeda, Tatsuhiko N. and Fujii, Keisuke}, + journal={arXiv preprint arXiv:2410.14243}, year={2026}, + eprint={2410.14243}, archivePrefix={arXiv}, primaryClass={quant-ph} +} +@article{malezic2026schemes, + title={Reducing the Gate Count with Efficient Trotter--Suzuki Schemes}, + author={Male\v{z}i\v{c}, Marko and Ostmeyer, Johann}, + journal={arXiv preprint arXiv:2602.21145}, year={2026}, + eprint={2602.21145}, archivePrefix={arXiv}, primaryClass={quant-ph} +} +@article{tate2026ordering, + title={An Analysis of Commutation-Based Trotter Ordering Strategies on Heisenberg-Style Hamiltonians}, + author={Tate, Reuben and Aktar, Shamminuj and Eidenbenz, Stephan}, + journal={arXiv preprint arXiv:2604.23138}, year={2026}, + eprint={2604.23138}, archivePrefix={arXiv}, primaryClass={quant-ph} +} +@article{maxwell2026practical, + title={Practical Estimation of Trotter Error for Hamiltonian Simulation}, + author={Maxwell, William and Casares, Pablo A. M. and Lang, Robert A. and Fomichev, Stepan and Arrazola, Juan Miguel and Jahangiri, Soran and Asadi, Ali and Nunez Meneses, Luis Alfredo and Germain, Thomas and Motlagh, Danial}, + journal={arXiv preprint arXiv:2606.30738}, year={2026}, + eprint={2606.30738}, archivePrefix={arXiv}, primaryClass={quant-ph} +} +@article{wang2026hncc, + title={Trotter Error Compensation with Polylogarithmic Precision and Nested-Commutator Scaling without Ancillas}, + author={Wang, Xinzhao and Zhou, Shuo and Wang, Ziruo and Zeng, Pei and Sun, Jinzhao and Zhao, Qi and Gur, Tom and Li, Tongyang}, + journal={arXiv preprint arXiv:2607.11856}, year={2026}, + eprint={2607.11856}, archivePrefix={arXiv}, primaryClass={quant-ph} +} +@article{suzuki1991general, + title={General Theory of Fractal Path Integrals with Applications to Many-Body Theories and Statistical Physics}, + author={Suzuki, Masuo}, journal={Journal of Mathematical Physics}, + volume={32}, pages={400--407}, year={1991}, doi={10.1063/1.529425} +} +@book{reutenauer1993free, + title={Free Lie Algebras}, author={Reutenauer, Christophe}, + publisher={Oxford University Press}, year={1993}, + doi={10.1093/oso/9780198536796.001.0001}, isbn={9780198536796} +} +@phdthesis{gottesman1997stabilizer, + title={Stabilizer Codes and Quantum Error Correction}, author={Gottesman, Daniel}, + school={California Institute of Technology}, year={1997}, eprint={quant-ph/9705052}, archivePrefix={arXiv} +} diff --git a/tracks/qcs/solutions/WangTheoPhys/issue128/docs/manuscript/scripts/generate_figures.py b/tracks/qcs/solutions/WangTheoPhys/issue128/docs/manuscript/scripts/generate_figures.py new file mode 100644 index 000000000..1faef3b9b --- /dev/null +++ b/tracks/qcs/solutions/WangTheoPhys/issue128/docs/manuscript/scripts/generate_figures.py @@ -0,0 +1,91 @@ +#!/usr/bin/env python3 +import json +from pathlib import Path + +import matplotlib.pyplot as plt +import numpy as np + +ROOT = Path(__file__).resolve().parents[3] +OUT = Path(__file__).resolve().parents[1] / "figures" +OUT.mkdir(parents=True, exist_ok=True) + +with (ROOT / "artifacts/issue128-summary.json").open() as handle: + summary = json.load(handle) + +plt.rcParams.update({ + "font.size": 9, + "axes.titlesize": 10, + "axes.labelsize": 9, + "legend.fontsize": 8, + "pdf.fonttype": 42, + "ps.fonttype": 42, +}) +blue, green, orange, red = "#2455A4", "#2A7F62", "#D97706", "#B42318" + +labels = ["Steps", "Merged groups", "Bond propagators", "CNOT upper"] +baseline = np.array([393, 11791, 848952, 2546856], dtype=float) +candidate = np.array([97, 2911, 209592, 628776], dtype=float) +x = np.arange(len(labels)) +fig, ax = plt.subplots(figsize=(6.6, 3.1)) +ax.bar(x - 0.19, np.ones(4), 0.38, label="Pinned baseline", color=blue) +ax.bar(x + 0.19, candidate / baseline, 0.38, label="Certified result", color=green) +ax.set_xticks(x, labels) +ax.set_ylabel("Resource normalized to baseline") +ax.set_ylim(0, 1.08) +for i, ratio in enumerate(candidate / baseline): + ax.text(i + 0.19, ratio + 0.025, f"{ratio:.3f}", ha="center", va="bottom") +ax.legend(frameon=False, ncol=2, loc="upper right") +ax.grid(axis="y", alpha=0.25) +fig.tight_layout() +fig.savefig(OUT / "resources.pdf", bbox_inches="tight") +plt.close(fig) + +ledger = summary["error_ledger"] +names = ["D4", "D5", "D6", "D7", "D8+"] +values = [float(ledger[k]["decimal_outward"].replace("e", "E")) for k in ["degree4", "degree5", "degree6", "degree7", "tail"]] +fig, ax = plt.subplots(figsize=(6.6, 3.15)) +bars = ax.bar(names, np.array(values) * 1e7, color=[blue, green, orange, "#7A5195", red]) +ax.axhline(10.0, color="black", linestyle="--", linewidth=1, label=r"total tolerance $10^{-6}$") +ax.set_ylabel(r"Contribution ($10^{-7}$)") +ax.set_title(r"Certified right-generator ledger at $r=97$") +for bar, value in zip(bars, values): + ax.text(bar.get_x() + bar.get_width()/2, bar.get_height() + 0.12, f"{value:.2e}", ha="center", fontsize=8) +ax.legend(frameon=False) +ax.grid(axis="y", alpha=0.25) +fig.tight_layout() +fig.savefig(OUT / "error_ledger.pdf", bbox_inches="tight") +plt.close(fig) + +direct, grouped = 20.160968407335066, 6.472926505087888 +fig, ax = plt.subplots(figsize=(5.2, 3.0)) +bars = ax.bar(["Termwise Pauli $\\ell_1$", "Certified anticommuting groups"], [direct, grouped], color=[orange, green]) +ax.set_ylabel("D4 translation-cell norm bound") +ax.set_title("Norm-last structural reduction") +ax.text(0.5, 0.82, f"{direct/grouped:.6f}x tighter", transform=ax.transAxes, ha="center", color=blue, fontweight="bold") +for bar, value in zip(bars, [direct, grouped]): + ax.text(bar.get_x()+bar.get_width()/2, value+0.35, f"{value:.6f}", ha="center", fontsize=8) +ax.set_ylim(0, 23) +ax.grid(axis="y", alpha=0.25) +fig.tight_layout() +fig.savefig(OUT / "d4_norm.pdf", bbox_inches="tight") +plt.close(fig) + +five_names = ["D4", "D5", "D6", "D7", "D8+"] +five_values = np.array([1.2590841639313686e-6, 4.1149153826738221e-7, 5.8406421306735928e-7, 8.7150631476060649e-8, 9.0076387927122002e-7]) +fig, ax = plt.subplots(figsize=(6.6, 3.45)) +bottom = 0.0 +colors = [blue, green, orange, "#7A5195", red] +for name, value, color in zip(five_names, five_values, colors): + ax.bar(["$r=78$ conditional audit"], [value*1e6], bottom=bottom, label=name, color=color) + bottom += value*1e6 +ax.axhline(1.0, color="black", linestyle="--", linewidth=1.2, label=r"target $10^{-6}$") +ax.set_ylabel(r"Error upper-bound ledger ($10^{-6}$)") +ax.set_title("Fivefold feasibility gap: not a global certificate", pad=10) +ax.text(0, bottom-0.10, f"total = {bottom:.3f} x target", ha="center", va="top", color="white", fontweight="bold") +ax.set_ylim(0, 3.65) +ax.legend(frameon=False, ncol=1, loc="upper left", bbox_to_anchor=(1.01, 1.0), borderaxespad=0) +fig.tight_layout() +fig.savefig(OUT / "fivefold_gap.pdf", bbox_inches="tight") +plt.close(fig) + +print(f"figures={len(list(OUT.glob('*.pdf')))}") diff --git a/tracks/qcs/solutions/WangTheoPhys/issue128/docs/manuscript/scripts/validate_claims.py b/tracks/qcs/solutions/WangTheoPhys/issue128/docs/manuscript/scripts/validate_claims.py new file mode 100644 index 000000000..6d73ac1e4 --- /dev/null +++ b/tracks/qcs/solutions/WangTheoPhys/issue128/docs/manuscript/scripts/validate_claims.py @@ -0,0 +1,38 @@ +#!/usr/bin/env python3 +from fractions import Fraction +import json +from pathlib import Path + +ROOT = Path(__file__).resolve().parents[3] + +with (ROOT / "artifacts/issue128-summary.json").open() as handle: + summary = json.load(handle) +with (ROOT / "certificates/issue128-certificate.json").open() as handle: + certificate = json.load(handle) + +baseline = certificate["published_baseline"] +candidate = certificate["candidate"] +resources = certificate["claimed_resources"] + +assert baseline["steps"] == 393 +assert candidate["steps"] == 97 +assert resources["published_group_exponentials"] == 11791 +assert resources["candidate_group_exponentials"] == 2911 +assert resources["published_bond_propagators"] == 848952 +assert resources["candidate_bond_propagators"] == 209592 +assert resources["published_cnot_upper"] == 2546856 +assert resources["candidate_cnot_upper"] == 628776 +assert Fraction(11791, 2911) > 4 +assert summary["improvement"]["exact_ratio"] == [11791, 2911] + +tolerance = Fraction(*certificate["benchmark"]["tolerance"]) +accepted = Fraction(*candidate["global_error_upper"]) +rejected = Fraction(*candidate["previous_step_error_upper"]) +assert accepted < tolerance +assert rejected > tolerance +assert candidate["d4_certificate"]["term_count"] == 75324 +assert candidate["d4_certificate"]["group_count"] == 7576 +assert candidate["d4_certificate"]["max_group_size"] == 10 +assert summary["fivefold_followup"]["status"] == "not_certified" + +print("headline_claims=PASS") diff --git a/tracks/qcs/solutions/WangTheoPhys/issue128/docs/manuscript/sections/abstract.tex b/tracks/qcs/solutions/WangTheoPhys/issue128/docs/manuscript/sections/abstract.tex new file mode 100644 index 000000000..8291a4820 --- /dev/null +++ b/tracks/qcs/solutions/WangTheoPhys/issue128/docs/manuscript/sections/abstract.tex @@ -0,0 +1,34 @@ +\begin{abstract} +Rigorous product-formula resource estimates often lose most of their useful +model structure before the final norm is evaluated. We introduce a +proof-carrying, model-aware compiler that delays norm inequalities while +translating a product formula through four intermediate representations: +sparse free associative words, homogeneous free-Lie defects, concrete +symplectic Pauli ledgers, and exact rational norm certificates. The compiler +combines identical translated Pauli strings before taking norms and replaces +termwise Pauli-$\ell_1$ estimates by independently verified +pairwise-anticommuting groups whenever possible. A finite right-logarithmic +generator expansion and an outward-rational geometric tail then convert the +local certificate into a nonasymptotic global error bound and an integer gate +count. + +For the periodic $12\times12$ spin-$1/2$ isotropic Heisenberg model at +$T=1$ and global operator-norm tolerance $10^{-6}$, we compare the same four +matching fragments, the same 31-stage five-copy fourth-order Suzuki formula, +and the same compilation cost on both sides. A pinned interval instantiation +of the Childs--Su--Tran--Wiebe--Zhu high-order commutator theorem requires 393 +steps and 11,791 merged group exponentials. Our certificate accepts 97 steps +and rejects 96, requiring 2,911 groups. The exact improvement is +$11791/2911=4.050498110614909\ldots$, a 75.3117\% reduction. The leading D4 +sidecar contains 75,324 canonical Pauli terms partitioned into 7,576 verified +anticommuting groups. The search procedure is outside the trusted computing +base: a smaller verifier rechecks coverage, every anticommutation relation, +outward square-root bounds, the finite-step ledger, the adjacent integer +boundary, and all resource arithmetic. An exact D5 side study and a +quantitative fivefold feasibility audit are reported separately; no 78-step +global certificate is claimed. +\end{abstract} + +\noindent\textbf{Keywords:} Hamiltonian simulation; Trotter--Suzuki product +formulas; rigorous error bounds; Pauli algebra; computer-assisted proof; +quantum resource estimation. diff --git a/tracks/qcs/solutions/WangTheoPhys/issue128/docs/manuscript/sections/appendices.tex b/tracks/qcs/solutions/WangTheoPhys/issue128/docs/manuscript/sections/appendices.tex new file mode 100644 index 000000000..02c430b86 --- /dev/null +++ b/tracks/qcs/solutions/WangTheoPhys/issue128/docs/manuscript/sections/appendices.tex @@ -0,0 +1,195 @@ +\section{Right-Generator Expansion Details} +\label{app:right-generator} + +This appendix records the finite-order identity used by the certificate. Let +\begin{equation} + Z(t)=\log S_4(t)=tA+t^5B+t^7C+O(t^9). +\end{equation} +Using +\begin{equation} + \operatorname{dexp}_{Z}(Z')= + Z'+\frac12[Z,Z']+\frac16[Z,[Z,Z']] + +\frac1{24}[Z,[Z,[Z,Z']]]+\cdots, +\end{equation} +and collecting powers through $t^7$ gives +\begin{align} + S_4'(t)S_4(t)^{-1}-A + ={}&5t^4B+2t^5[A,B] + +t^6\left(7C+\frac23[A,[A,B]]\right)\nonumber\\ + &+t^7\left(3[A,C]+\frac16[A,[A,[A,B]]]\right) + +R_{\geq8}(t). + \label{eq:exact-right-expansion} +\end{align} +For example, the $t^5[A,B]$ coefficient comes from +$\frac12[tA,5t^4B]+\frac12[t^5B,A]=2t^5[A,B]$. +The later coefficients follow by the same homogeneous bookkeeping. The +implementation constructs the series in the word algebra and regression-tests +\cref{eq:exact-right-expansion}; it does not substitute the printed identity +as an unchecked constant table. + +The global contribution of a monomial $t^jD_j$ follows from +\cref{eq:global-duhamel}: +\begin{equation} + r\int_0^{T/r}s^j\opnorm{D_j}\,ds + =\frac{T^{j+1}\opnorm{D_j}}{(j+1)r^j}. + \label{eq:monomial-global} +\end{equation} +This explains the degree labels in the result ledger and provides a direct +dimensional check on the inverse powers of $r$. + +\section{Local Heisenberg Pauli Audit} +\label{app:local-pauli} + +The single-bond lemma can be checked without symbolic software. Let +$Q_1=XX$, $Q_2=YY$, and $Q_3=ZZ$. For a phase-free two-qubit Pauli +$P=P_1\otimes P_2$, the number of $Q_j$ that anticommute with $P$ is either +zero or two. One compact enumeration is shown in \cref{tab:local-audit}; rows +and columns give the one-qubit factors. + +\begin{table}[ht] +\centering +\caption{Number of anticommuting terms among $XX,YY,ZZ$ for every local +two-qubit Pauli.} +\label{tab:local-audit} +\begin{tabular}{@{}c|rrrr@{}} +\toprule +$P_1\backslash P_2$ & $I$ & $X$ & $Y$ & $Z$ \\ +\midrule +$I$ & 0 & 2 & 2 & 2 \\ +$X$ & 2 & 0 & 2 & 2 \\ +$Y$ & 2 & 2 & 0 & 2 \\ +$Z$ & 2 & 2 & 2 & 0 \\ +\bottomrule +\end{tabular} +\end{table} + +If the count is zero, $[h,P]=0$. If it is two, each contributing Hamiltonian +coefficient has magnitude $1/4$ and the Pauli commutator supplies magnitude +two, giving two output coefficients of magnitude $1/2$. Their +Pauli-$\ell_1$ sum is one. Hence +\begin{equation} + \paulinorm{[h,P]}\leq1=\paulinorm{P} +\end{equation} +for a unit Pauli input, and linearity extends the stated growth majorant. The +generic bound obtained by summing the three Hamiltonian Paulis independently +would give $3/2$; the exact parity enumeration removes the impossible third +contribution. + +\section{Verifier Pseudocode} +\label{app:verifier} + +\begin{algorithm}[ht] +\caption{Independent verification of the principal certificate} +\label{alg:verifier} +\begin{algorithmic}[1] +\Require main certificate $C$ and sidecar directory +\State validate schema version and required benchmark fields +\State recompute SHA-256 for every referenced sidecar +\State verify formula, normalization, $T$, $\eps$, and cost-model identities +\State load the D4 coefficient ledger and declared group partition +\State verify exact coverage: no missing, duplicate, or unknown Pauli key +\For{every group $G$} + \For{every unordered pair $(P,Q)\subseteq G$} + \State require symplectic parity$(P,Q)=1$ + \EndFor + \State recompute the rational squared-weight sum + \State require declared upper root squared $\geq$ recomputed sum +\EndFor +\State sum all group roots and compare to the D4 cell enclosure +\State verify D5--D7 and tail preconditions and contributions +\State sum the accepted-step global rational; require result $\leq\eps$ +\State sum the previous-step rational; require result $>\eps$ +\State recompute $G(r)$, bond propagators, CNOT totals, and exact ratio +\State return a structured success record; fail closed on any mismatch +\end{algorithmic} +\end{algorithm} + +Deep mode inserts regeneration of the baseline center scan, complete formula +logarithm, DSW defects, Pauli coefficient map, and higher-degree majorants +before the certificate comparisons. Importantly, both modes fail closed: +unknown schema versions, absent files, extra group members, and invalid +interval orientation are errors rather than warnings. + +\section{Certificate Schema Sketch} +\label{app:schema} + +\begin{longtable}{@{}p{0.29\textwidth}p{0.19\textwidth}p{0.44\textwidth}@{}} +\caption{Selected schema-v3 fields and verification obligations.} +\label{tab:schema}\\ +\toprule +Field & Type & Obligation \\ +\midrule +\endfirsthead +\toprule +Field & Type & Obligation \\ +\midrule +\endhead +\path{benchmark} & object & Exact model, normalization, size, time, +tolerance, and primary metric match the claim \\ +\path{published_baseline.site_density_upper} & integer pair & Denominator is +positive and deep regeneration is enclosed \\ +\path{candidate.d4_certificate.path} & relative path & Resolves inside the +bundle; no path traversal \\ +\path{candidate.d4_certificate.sha256} & 64-hex digest & Equals the bytes of +the resolved sidecar \\ +\path{candidate.contributions} & rational pairs & Each pair has positive +denominator and agrees with recomputed bounds \\ +\path{candidate.global_error_upper} & rational pair & Equals or dominates +the contribution sum and is below tolerance \\ +\path{candidate.previous_step_error_upper} & rational pair & Correctly +evaluated at $r-1$ and exceeds tolerance \\ +\path{cost_model} & integers/rule & Resource identities are evaluated without +floating point \\ +\path{claimed_resources} & integer object & Every total is reconstructed from +the step count and cost model \\ +\path{claims.exact_improvement_ratio} & integer pair & Equals the declared +baseline and candidate group counts \\ +\bottomrule +\end{longtable} + +\section{Additional Interpretation of the Improvement} +\label{app:interpretation} + +Several ratios appear in the analysis and should not be conflated: +\begin{align} +\text{D4 norm tightening} + &=\frac{20.160968407335066}{6.472926505087888} + =3.114660484942633\ldots,\\ +\text{step ratio} + &=\frac{393}{97}=4.051546391752577\ldots,\\ +\text{compiled resource ratio} + &=\frac{30\cdot393+1}{30\cdot97+1} + =\frac{11791}{2911} + =4.050498110614909\ldots. +\end{align} +The step and resource ratios differ because the first and last fragment groups +merge to give the additive one in \cref{eq:merged-groups}. The norm ratio is +not expected to equal either: a fourth-order global error scales roughly as +$r^{-4}$ only in a leading asymptotic regime, while the actual accepted point +contains D5--D7, a tail, and integer rounding. + +The improvement relative to the challenge's twofold target is +\begin{equation} + \frac{(11791/2911)}{2}=2.025249055307454\ldots. +\end{equation} +This means the achieved multiplier is just over twice the requested +multiplier; it does not mean an eightfold resource reduction. + +\section{Checklist for Independent Review} +\label{app:checklist} + +A reviewer can audit the result in increasing order of cost: +\begin{enumerate} +\item Read the benchmark tuple and confirm that candidate and control use the +same Hamiltonian, formula, tolerance, and cost model. +\item Run the manuscript headline validator and the package checksum check. +\item Run the fast verifier and inspect its structured output. +\item Confirm the exact integer map from 393/97 steps to all three resources. +\item Run the default unit suite, including mutation and local-Pauli tests. +\item Run D5 \code{--verify-only} to confirm the follow-up is separate and +marked noncertified at 78 steps. +\item If computational resources permit, run the deep verifier to regenerate +the formula-specific algebra and baseline center scan. +\item Rebuild this PDF and visually inspect figures, tables, citations, and +page boundaries; PDF appearance is a communication check, not a proof check. +\end{enumerate} diff --git a/tracks/qcs/solutions/WangTheoPhys/issue128/docs/manuscript/sections/baseline.tex b/tracks/qcs/solutions/WangTheoPhys/issue128/docs/manuscript/sections/baseline.tex new file mode 100644 index 000000000..4015aeb70 --- /dev/null +++ b/tracks/qcs/solutions/WangTheoPhys/issue128/docs/manuscript/sections/baseline.tex @@ -0,0 +1,81 @@ +\section{Reconstructing the Published-Theorem Baseline} +\label{sec:baseline} + +The phrase ``published baseline'' can obscure two logically distinct objects. +The first is the high-order commutator theorem published by Childs et al. +\cite{childs2021trotter}; the second is our fixed, machine-evaluated +instantiation of that theorem. The number 393 is not a table entry quoted +from the 2021 paper. It is obtained by applying the published framework to +\cref{eq:hamiltonian,eq:benchmark,eq:suzuki4} with a specified norming and +interval procedure. We call it the \emph{pinned control}. + +\subsection{Expansion-first local norming} + +The high-order theorem represents product-formula error by partial sums and +nested commutators. For the pinned control, every partial matching sum is +expanded into local Heisenberg bonds, zero commutators are removed, and each +surviving concrete Pauli expansion is bounded by its Pauli-$\ell_1$ norm. +The resulting site-density upper bound is stored as an exact rational number +whose outward decimal is +\begin{equation} + \alpha_{\mathrm{base}}\le164.9759169527440. +\end{equation} +The calculation contains 8,316 theorem terms and 1,024 expanded commutator +keys. All algebraic product-formula coefficients are enclosed by rational +intervals; no rounded decimal is used as the source of the result. + +The theorem admits a discrete family of center choices controlling how the +high-order integral representation is partitioned. We evaluate all 31 +admissible centers. Center 20 gives the smallest certified control, and +inverting its global bound at tolerance $10^{-6}$ yields +\begin{equation} + r_{\mathrm{base}}=393,\qquad G(r_{\mathrm{base}})=11{,}791. +\end{equation} +The main certificate stores both the exact site-density rational and the +chosen center. A deep verification mode regenerates the center scan rather +than trusting these stored labels. + +\subsection{Where the baseline loses information} + +The baseline is rigorous, general, and already exploits exact commutation. +Its looseness in this instance arises from the order of operations. Once an +abstract contribution is replaced by an independent positive norm, later +knowledge cannot recover cancellation between that contribution and another +one. Four losses are particularly important: + +\begin{enumerate} +\item distinct free words can map to the same nested-commutator or Pauli + contribution with opposite signs; +\item distinct translated local clusters can share a canonical translation + representative; +\item different commutator paths can produce exactly the same Pauli string; +\item different surviving Pauli strings can anticommute, causing cross terms + to vanish when the group norm is squared. +\end{enumerate} + +The proposed compiler postpones the irreversible inequality until these +relations have been exposed. Importantly, it does not treat the baseline as +incorrect. It constructs a stronger proof for the same circuit and then +verifies both sides under a common benchmark and cost model. + +\begin{table}[t] +\centering +\caption{Logical separation between the published theorem and the pinned +finite-instance control.} +\label{tab:baseline-separation} +\begin{tabularx}{\textwidth}{@{}lXX@{}} +\toprule + & Published theory & Pinned control used here \\ +\midrule +Source & Childs et al. commutator-scaling framework & Exact interval +instantiation for the fixed Heisenberg benchmark \\ +Scope & General Hamiltonian fragments and formula order & $L=12$, four +matchings, five-copy $S_4$, $T=1$, $\eps=10^{-6}$ \\ +Norming & Theorem permits commutator-sensitive evaluation & Expanded local +Pauli-$\ell_1$ procedure fixed by the certificate \\ +Finite number & No published 393-step table entry & 31 centers scanned; +center 20 yields 393 steps \\ +Role & Mathematical foundation & Auditable comparison control \\ +\bottomrule +\end{tabularx} +\end{table} diff --git a/tracks/qcs/solutions/WangTheoPhys/issue128/docs/manuscript/sections/compiler.tex b/tracks/qcs/solutions/WangTheoPhys/issue128/docs/manuscript/sections/compiler.tex new file mode 100644 index 000000000..8e0e382f2 --- /dev/null +++ b/tracks/qcs/solutions/WangTheoPhys/issue128/docs/manuscript/sections/compiler.tex @@ -0,0 +1,120 @@ +\section{A Proof-Carrying Compiler for Error Bounds} +\label{sec:compiler} + +\subsection{Intermediate representations} + +The compiler is organized around four intermediate representations (IRs). +The \emph{word IR} stores sparse homogeneous polynomials in noncommuting +letters $A_1,\ldots,A_4$. The \emph{Lie IR} stores homogeneous logarithmic +defects as rational linear combinations of right-nested commutators. The +\emph{Pauli IR} stores canonical finite-support symplectic Pauli strings with +exact coefficients per translation cell. The \emph{certificate IR} stores +partitions, rational norm enclosures, finite-step contributions, tail data, +hash bindings, and resource arithmetic. + +\begin{figure}[t] +\centering +\resizebox{\textwidth}{!}{% +\begin{tikzpicture}[ + node distance=5mm and 6mm, + box/.style={draw,rounded corners,align=center,minimum height=8mm,minimum width=25mm,fill=blue!4}, + certbox/.style={box,fill=green!8}, + arr/.style={-{Stealth[length=2mm]},thick} +] +\node[box] (input) {Hamiltonian, fragments,\\formula, $T,\eps$, cost}; +\node[box,right=of input] (word) {Free-word IR\\formal logarithm}; +\node[box,right=of word] (lie) {Free-Lie IR\\DSW projection}; +\node[box,right=of lie] (pauli) {Pauli IR\\translation quotient}; +\node[certbox,right=of pauli] (cert) {Certificate IR\\norm and tail}; +\node[certbox,below=of cert] (output) {$r_\star$, adjacent boundary,\\integer resources}; +\draw[arr] (input)--(word); +\draw[arr] (word)--(lie); +\draw[arr] (lie)--(pauli); +\draw[arr] (pauli)--(cert); +\draw[arr] (cert)--(output); +\end{tikzpicture} +} +\caption{Compiler pipeline. Algebraic structure is retained until the Pauli +IR; irreversible norm inequalities enter only in the certificate IR.} +\label{fig:pipeline} +\end{figure} + +Each lowering pass preserves a semantic invariant. Word multiplication +preserves the truncated product series. Formal logarithms preserve the +truncated exponential identity. DSW projection preserves the homogeneous +Lie element. Pauli evaluation is a representation homomorphism. Translation +canonicalization preserves the sum per cell. Exact aggregation preserves the +represented operator. Finally, grouped norming replaces equality by a +verified upper bound. + +\subsection{End-to-end algorithm} + +\begin{algorithm}[t] +\caption{Compile a product formula into a resource certificate} +\label{alg:compiler} +\begin{algorithmic}[1] +\Require $(H,\mathcal F,S,T,\eps,\mathcal C)$ and truncation policy +\Ensure Certificate accepted by an independent verifier +\State parse all stage coefficients into exact algebraic or outward intervals +\State $P(t)\gets$ truncated free-word expansion of the complete stage product +\State $L(t)\gets\log P(t)$ in the sparse free associative algebra +\State verify order conditions and time-reversal parity through target degree +\For{$d\in\{5,7\}$} + \State $E_d\gets$ DSW projection of the homogeneous component $L_d$ + \State $Q_d\gets$ evaluate $E_d$ in the concrete local Pauli representation + \State quotient $Q_d$ by translation and combine identical Pauli strings +\EndFor +\State discover weighted pairwise-anticommuting groups for the leading ledger +\State emit full group membership and outward rational square-root bounds +\State convert $E_5,E_7$ to right-generator terms D4--D7 +\State close all remaining terms with a rational locality majorant and tail +\State find the smallest integer $r_\star$ whose certified bound is at most $\eps$ +\State compile $r_\star$ to groups, bond propagators, and CNOT upper bound +\State emit hashes, provenance, and adjacent-step evidence +\end{algorithmic} +\end{algorithm} + +The algorithm intentionally separates \emph{discovery} from \emph{trust}. +The formal-series generator and grouping search can be large and optimized for +throughput. Their output is not accepted because the code ran successfully; +it is accepted only after a second path rechecks the certificate. In +particular, no claim depends on the greedy grouping being optimal. Any valid +partition gives a sound bound, and a better heuristic can improve the bound +without expanding the trusted computing base. + +\subsection{What is improved at each pass} + +\begin{table}[t] +\centering +\caption{Compiler passes and the information they preserve.} +\label{tab:passes} +\begin{tabularx}{\textwidth}{@{}p{0.19\textwidth}p{0.26\textwidth}Xp{0.18\textwidth}@{}} +\toprule +Pass & Conventional relaxation & Replacement & Verified invariant \\ +\midrule +Complete formula log & Bound stages or partial sums separately & Multiply all +stages and take the formal log before norming & Truncated series identity and +order conditions \\ +Lie projection & Treat free words as unrelated monomials & Project homogeneous +log defects into free-Lie elements & DSW identity at fixed degree \\ +Concrete evaluation & Count generic nonzero commutators & Evaluate exact local +Pauli products and phases & Symplectic multiplication and support \\ +Translation quotient & Count translated clusters separately & Select a +canonical orbit representative & Translation-equivalent configurations merge \\ +Coefficient aggregation & Take absolute values by derivation path & Sum exact +coefficients of identical Paulis & Operator represented by sparse ledger \\ +Anticommuting grouping & Apply termwise Pauli-$\ell_1$ & Use exact group +$\ell_2$ identities & Coverage and pairwise anticommutation \\ +Finite-step closure & Use only leading order & Retain D4--D7 and rational tail +& Global error upper bound \\ +Adjacent integer search & Round a continuous estimate & Verify $r$ and $r-1$ +exactly & Smallest integer accepted by this bound \\ +\bottomrule +\end{tabularx} +\end{table} + +This organization also answers how the method should be extended. A new +Hamiltonian needs a representation backend and local-support rules; a new +formula needs a stage generator and exact coefficient enclosure; a new norm +strategy needs a certificate schema and verifier. The higher layers need not +be rewritten if their interfaces and invariants remain unchanged. diff --git a/tracks/qcs/solutions/WangTheoPhys/issue128/docs/manuscript/sections/conclusion.tex b/tracks/qcs/solutions/WangTheoPhys/issue128/docs/manuscript/sections/conclusion.tex new file mode 100644 index 000000000..26858123e --- /dev/null +++ b/tracks/qcs/solutions/WangTheoPhys/issue128/docs/manuscript/sections/conclusion.tex @@ -0,0 +1,26 @@ +\section{Conclusion} +\label{sec:conclusion} + +We have presented a proof-carrying, model-aware compiler for rigorous +product-formula resource bounds. Its central design rule is simple: preserve +algebraic structure until all formula-specific, model-specific, translational, +and anticommuting relations have been exposed; apply irreversible norm +inequalities only afterward; and emit enough evidence for an independent +program to recheck the result. + +For the periodic $12\times12$ isotropic Heisenberg benchmark, this procedure +certifies the original five-copy fourth-order Suzuki circuit at 97 steps. The +pinned general-theorem instantiation requires 393 steps under the common +comparison protocol. After deterministic stage merging and bond +compilation, the corresponding group, bond, and CNOT resources fall by the +exact factor $11791/2911=4.050498110614909\ldots$. The result includes the +rejected 96-step neighbor, D4--D7 contributions, a rational high-degree tail, +full D4 anticommuting membership, resource arithmetic, and hash bindings. + +The broader lesson is not that one Hamiltonian has exhausted the possibilities +of Trotter simulation. It is that a general error theorem and a concrete gate +estimate should be connected by a transparent compiler rather than a chain of +unrecorded inequalities. The current artifact occupies that middle layer. A +fivefold threshold remains a well-specified research target, but it will be +claimed only after the missing D4, D8, and tail obligations close in the same +machine-checkable form. diff --git a/tracks/qcs/solutions/WangTheoPhys/issue128/docs/manuscript/sections/finite_step.tex b/tracks/qcs/solutions/WangTheoPhys/issue128/docs/manuscript/sections/finite_step.tex new file mode 100644 index 000000000..5f205cf64 --- /dev/null +++ b/tracks/qcs/solutions/WangTheoPhys/issue128/docs/manuscript/sections/finite_step.tex @@ -0,0 +1,113 @@ +\section{Finite-Step Right-Generator Certificate} +\label{sec:finite-step} + +An asymptotic statement $O(t^5)$ is insufficient at the target tolerance. +The certified step count is obtained from a finite-$t$ right-generator bound, +including explicit subleading terms and a rational tail. + +\subsection{From logarithmic defect to right generator} + +Let $S(t)$ denote one formula step and define its right generator +\begin{equation} + K(t)=S'(t)S(t)^{-1}. + \label{eq:right-generator} +\end{equation} +For $Z(t)=\log S(t)$, differentiation of the exponential gives +\begin{equation} + K(t)=\operatorname{dexp}_{Z(t)}(Z'(t)) + =\sum_{n\geq0}\frac{1}{(n+1)!}\ad_{Z(t)}^n Z'(t). + \label{eq:dexp} +\end{equation} +Substituting \cref{eq:symmetric-log} and collecting powers yields +\begin{equation} + K(t)-A=t^4D_4+t^5D_5+t^6D_6+t^7D_7+R_{\geq8}(t), + \label{eq:right-ledger} +\end{equation} +where, writing $B=E_5$ and $C=E_7$, the exact coefficients through degree +seven are +\begin{align} + D_4&=5B, & + D_5&=2[A,B],\\ + D_6&=7C+\tfrac23[A,[A,B]], & + D_7&=3[A,C]+\tfrac16[A,[A,[A,B]]]. +\end{align} +The implementation derives these coefficients from the truncated series +rather than trusting the printed display. The identity clarifies why an +exact leading logarithmic coefficient is not yet a global error certificate: +commutators generated by the differential of the exponential contribute at +the next powers. + +\subsection{Duhamel conversion to global error} + +Let $U(t)=e^{tA}$ in anti-Hermitian convention. Variation of constants gives +the one-step estimate +\begin{equation} + \opnorm{S(\delta)-U(\delta)} + \leq \int_0^\delta \opnorm{K(s)-A}\,ds. + \label{eq:one-step-duhamel} +\end{equation} +Unitary telescoping over $r$ equal steps, with $\delta=T/r$, then yields +\begin{equation} + \opnorm{S(T/r)^r-U(T)} + \leq r\int_0^{T/r}\opnorm{K(s)-A}\,ds. + \label{eq:global-duhamel} +\end{equation} +Combining \cref{eq:right-ledger,eq:global-duhamel}, a certificate with norm +upper bounds $b_j\geq\opnorm{D_j}$ and a nonnegative tail majorant $\rho(s)$ +establishes +\begin{equation} + \mathcal E(r)= + r\left(\sum_{j=4}^{7}\frac{b_j}{j+1} + \left(\frac{T}{r}\right)^{j+1} + +\int_0^{T/r}\rho(s)\,ds\right). + \label{eq:finite-step-bound} +\end{equation} +All quantities in the machine certificate are rationals. Decimal values in +this paper are outward-rounded summaries only. + +\subsection{Rational tail closure} + +The exact ledger is stopped after D7. Higher degrees are bounded by a +geometric majorant derived from the absolute stage coefficients and local +commutator growth. In abstract form, if the degree-$k$ coefficient is at most +$M q^{k-k_0}$ for $0\leq q s<1$, then +\begin{equation} + \rho(s)\leq \frac{M s^{k_0}}{1-qs}. + \label{eq:geometric-tail} +\end{equation} +The verifier checks the rational preconditions, including positivity of the +denominator on $[0,T/r]$, and integrates an outward upper enclosure. This is +deliberately more conservative than extrapolating the observed low-degree +coefficients. It ensures that no uncomputed term is silently discarded. + +\begin{theorem}[Soundness of an accepted finite-step certificate] +\label{thm:soundness} +Suppose the verifier accepts (i) interval enclosures for D4--D7, (ii) a valid +operator-norm certificate for each declared coefficient, (iii) a tail +majorant satisfying \cref{eq:geometric-tail} on $[0,T/r]$, and (iv) exact +rational evaluation $\mathcal E(r)\leq\eps$. Then +\begin{equation} + \opnorm{S(T/r)^r-e^{TA}}\leq\eps. +\end{equation} +\end{theorem} + +\begin{proof} +The accepted coefficient and tail checks bound +$\opnorm{K(s)-A}$ term by term on the complete integration interval. +Integrating preserves the inequality. Equation~\eqref{eq:global-duhamel} +then bounds the global error, and the exact rational comparison with $\eps$ +completes the proof. +\end{proof} + +\subsection{Adjacent-step minimality under the certified bound} + +The search evaluates integers, not a rounded real-valued estimate. It returns +the least $r$ accepted by \cref{eq:finite-step-bound} and emits the rejected +neighbor. For this benchmark, +\begin{align} + \mathcal E(97)&\leq9.958938494314325\times10^{-7}<10^{-6},\\ + \mathcal E(96)&\leq1.050565873970784\times10^{-6}>10^{-6}. +\end{align} +The second inequality says that 96 is not certified by \emph{this bound}; it +does not prove that the physical simulation error at 96 steps exceeds the +tolerance. This distinction is retained throughout the paper. diff --git a/tracks/qcs/solutions/WangTheoPhys/issue128/docs/manuscript/sections/free_lie.tex b/tracks/qcs/solutions/WangTheoPhys/issue128/docs/manuscript/sections/free_lie.tex new file mode 100644 index 000000000..8cbe4612a --- /dev/null +++ b/tracks/qcs/solutions/WangTheoPhys/issue128/docs/manuscript/sections/free_lie.tex @@ -0,0 +1,106 @@ +\section{Complete-Formula Free-Lie Extraction} +\label{sec:free-lie} + +The first source of slack in a generic product-formula estimate is often not +the inequality used at the end, but the loss of cancellation before the +inequality is applied. We therefore expand the \emph{complete} 31-stage +formula as one formal object. This section describes the representation and +the identities checked by the implementation. + +\subsection{Sparse truncated words} + +Let $\mathbb{K}\langle A_1,\ldots,A_m\rangle$ denote the free associative +algebra over a coefficient field $\mathbb K$. A homogeneous word +$w=(j_1,\ldots,j_d)$ represents $A_{j_1}\cdots A_{j_d}$. Up to degree $q$, +we store a series as the sparse map +\begin{equation} + P(t)=1+\sum_{d=1}^{q}t^d\sum_{|w|=d}p_w w+O(t^{q+1}). + \label{eq:sparse-word-series} +\end{equation} +Multiplication is truncated convolution. An exponential stage is generated +by +\begin{equation} + \exp(a t A_j)=\sum_{k=0}^{q}\frac{(a t)^k}{k!}A_j^k+O(t^{q+1}), +\end{equation} +and all 31 stages are multiplied in their actual execution order. The +coefficients $a$ are held as outward rational intervals. Thus every stored +coefficient encloses the corresponding real algebraic coefficient, including +the fourth-order Suzuki coefficient involving $4^{1/3}$. + +For $X=P-1$, the formal logarithm is +\begin{equation} + \log P=\sum_{k=1}^{q}\frac{(-1)^{k+1}}{k}X^k+O(t^{q+1}). + \label{eq:formal-log} +\end{equation} +No matrices and no floating-point eigensolvers enter this computation. The +calculation is universal in the letters; only after the Lie defect is known do +we substitute the concrete Heisenberg fragments. + +\subsection{Order, symmetry, and the full-stage cancellation budget} + +Write the ideal generator as $A=\sum_{j=1}^{4}A_j$ and the logarithm of a +single formula step as +\begin{equation} + \log S_4(t)=tA+t^5E_5+t^7E_7+O(t^9). + \label{eq:symmetric-log} +\end{equation} +Fourth order requires the degree-two through degree-four defects to vanish. +Time symmetry, $S_4(-t)=S_4(t)^{-1}$, implies that the logarithm is odd, so its +even homogeneous components vanish. The compiler checks these statements +coefficient by coefficient in the word IR; it does not infer them merely from +the name of the formula. + +This global construction matters. If a bound is applied to each stage or to +each recursive subformula before the product is assembled, opposite +coefficients and Jacobi-related terms can no longer cancel. In our pipeline, +absolute values first appear only after the complete formula has been +projected, evaluated, translated, and aggregated. This ordering of +operations is the main methodological distinction from a direct +``expand--triangle-inequality'' instantiation of a general commutator theorem. + +\subsection{Dynkin--Specht--Wever projection} + +The homogeneous component $L_d$ of a logarithm of group-like series is a Lie +element. We use the Dynkin--Specht--Wever (DSW) map to expose that structure. +For a word $w=x_1\cdots x_d$, define +\begin{equation} + D(w)=[x_1,[x_2,\ldots,[x_{d-1},x_d]\ldots]]. +\end{equation} +The DSW theorem gives $D(L_d)=dL_d$ for homogeneous Lie elements +\citep{reutenauer1993free}. Consequently, +\begin{equation} + E_d=L_d=\frac{1}{d}\sum_{|w|=d}\ell_w D(w), + \label{eq:dsw} +\end{equation} +where $\ell_w$ are the word coefficients obtained from \cref{eq:formal-log}. +Equation~\eqref{eq:dsw} is used as a projection and as an independently +checkable identity: expanding its right-hand side back into words must recover +$L_d$ interval-wise. + +\begin{proposition}[Semantic preservation of the Lie lowering pass] +\label{prop:lie-pass} +Assume the input interval map encloses every coefficient of the degree-$d$ +homogeneous formal logarithm and the DSW reconstruction check succeeds. Then +the emitted nested-commutator interval map encloses the exact homogeneous Lie +defect $E_d$. +\end{proposition} + +\begin{proof} +The DSW map is linear over the coefficient field. Interval scalar +multiplication and addition are inclusion isotone, so applying $D/d$ to the +input enclosures contains its application to the exact coefficients. The DSW +identity fixes the exact image to $E_d$. The reconstruction check additionally +guards against word-order, sign, and truncation errors in the implementation. +\end{proof} + +\subsection{Why this is an algorithmic improvement} + +The free-Lie stage is not a new product formula: it is a better compiler for a +fixed formula. Its improvement is threefold. First, it preserves +cross-stage cancellations. Second, it replaces derivation histories by a +canonical operator expression. Third, it makes the later model substitution +sparse: nested commutators that vanish under the local Pauli representation +are removed exactly rather than charged by a worst-case commutator factor. +The cost is an offline symbolic calculation whose size grows rapidly with +degree. That cost is acceptable here because the certificate is generated +once and verified much more cheaply than it is discovered. diff --git a/tracks/qcs/solutions/WangTheoPhys/issue128/docs/manuscript/sections/introduction.tex b/tracks/qcs/solutions/WangTheoPhys/issue128/docs/manuscript/sections/introduction.tex new file mode 100644 index 000000000..3088e2559 --- /dev/null +++ b/tracks/qcs/solutions/WangTheoPhys/issue128/docs/manuscript/sections/introduction.tex @@ -0,0 +1,103 @@ +\section{Introduction} +\label{sec:introduction} + +Product formulas remain among the most transparent constructions for digital +Hamiltonian simulation. If $H=\sum_{\gamma=1}^{\Gamma}H_\gamma$, a product +formula replaces $\exp(-itH)$ by a sequence of exponentials of simpler +fragments $H_\gamma$. The method needs no block encoding, can preserve the +geometry of local interactions, and maps naturally to hardware with limited +connectivity. Its practical cost, however, is determined not merely by its +formal order but by the constant in a rigorous finite-time error bound. A +factor lost in this constant is amplified when the bound is inverted to select +an integer number of Trotter steps and then multiplied by every local +propagator in every step. + +The modern commutator-scaling theory of Childs, Su, Tran, Wiebe, and Zhu +provides a general foundation for such estimates +\cite{childs2021trotter}. It replaces purely norm-sum scaling by sums of +nested commutators and thereby exploits locality and exact commutation. This +framework is sufficiently general to cover arbitrary order and many +Hamiltonian classes, but a general theorem must remain conservative about the +concrete algebra of a fixed model. In a Pauli lattice, multiple formal +commutator paths may evaluate to the same operator with opposite signs; +translation-equivalent terms may repeat; and distinct surviving Paulis may +anticommute, so their sum has a Euclidean rather than taxicab norm. Applying a +triangle inequality before these facts become visible irreversibly discards +them. + +This paper studies the resulting ``last-mile'' problem: + +\begin{quote} +Given a fixed local Hamiltonian, fragmentation, product formula, simulation +time, error tolerance, and compilation model, how can one automatically +produce a tighter \emph{machine-checkable} worst-case resource bound without +changing the target quantum circuit family? +\end{quote} + +Our answer is to treat error analysis as compilation. The source program is +the Hamiltonian and product formula. The compiler lowers it through +successive algebraic intermediate representations, performs semantics- +preserving optimization passes, and emits a proof object. In contrast with a +normal circuit optimizer, the principal output is not a different sequence of +quantum gates. It is a smaller certified number of repetitions of the same +sequence. The design follows a proof-carrying principle: expensive symbolic +search may be heuristic, but an accepted result must carry all data required +for a simpler independent verifier to reconstruct the logical obligations. + +\subsection{Contributions} + +The main contributions are as follows. + +\begin{enumerate} +\item \textbf{A multi-representation error compiler.} We compute the + truncated logarithm of the complete product formula in a sparse free + associative algebra, project its homogeneous defects to free-Lie elements, + evaluate those elements in a concrete symplectic Pauli representation, and + finally emit exact rational certificate data. Each lowering step has an + explicit invariant and can be checked independently. + +\item \textbf{Norm-last concrete evaluation.} We canonicalize finite Pauli + configurations under lattice translation and combine identical Pauli + strings with exact coefficients before applying a final norm inequality. + This reverses the conventional expand-and-bound order and preserves + cancellations that are invisible at the level of abstract support counts. + +\item \textbf{Certified anticommuting norm compression.} We construct a + graph whose vertices are leading-defect Pauli terms and whose edges encode + anticommutation. An untrusted search partitions terms into pairwise- + anticommuting groups. The certificate records the full partition and + outward rational square-root bounds; the verifier rechecks exact coverage + and every edge used. For the principal instance, this lowers the D4 + translation-cell bound from 20.160968407335066 to 6.472926505087888. + +\item \textbf{A nonasymptotic right-generator closure.} We convert the + degree-five and degree-seven logarithmic defects into the right logarithmic + generator, retain degrees D4 through D7, use a Heisenberg-specific local + commutator lemma, and control all remaining degrees by a rational geometric + locality tail. Thus the resource result does not rely on leading-order + extrapolation. + +\item \textbf{An independently reproducible finite-instance result.} For a + $12\times12$ periodic Heisenberg lattice, the certificate accepts 97 steps + and rejects 96. Relative to a pinned 393-step instantiation of the + published high-order theorem, this reduces merged group exponentials from + 11,791 to 2,911. A fast verifier, deep regeneration mode, small-system + dense cross-check, mutation tests, and a SHA-256 manifest support the claim. +\end{enumerate} + +\subsection{Scope and nonclaims} + +The result is a finite-size constant-factor improvement, not a new asymptotic +Hamiltonian-simulation complexity. It applies to a uniform operator-norm +guarantee for a fixed deterministic product formula. Methods based on low- +energy inputs, average-case norms, multi-product formulas, linear combinations +of unitaries, extrapolation, or stochastic compensation solve different +resource problems and may achieve stronger precision scaling under their own +cost models. We compare these directions in \cref{sec:related} but do not +rank incomparable gate counts as a single global record. + +The exact D5 side study and the $r=78$ arithmetic indicate a possible +fivefold threshold. They do not establish it. Throughout this manuscript, +``certified'' refers only to statements accepted by the supplied rational +certificate and verifier. Numerical dense simulation is used solely as a +cross-check, and conditional research estimates are labeled explicitly. diff --git a/tracks/qcs/solutions/WangTheoPhys/issue128/docs/manuscript/sections/limitations.tex b/tracks/qcs/solutions/WangTheoPhys/issue128/docs/manuscript/sections/limitations.tex new file mode 100644 index 000000000..fd53de3ea --- /dev/null +++ b/tracks/qcs/solutions/WangTheoPhys/issue128/docs/manuscript/sections/limitations.tex @@ -0,0 +1,87 @@ +\section{Limitations and Research Roadmap} +\label{sec:limitations} + +The present evidence supports a precise but deliberately narrow conclusion. +This section identifies the boundaries and shows how each could be addressed +without weakening the current claim. + +\subsection{Finite instance rather than universal scaling} + +The numerical certificate is for the periodic $12\times12$ isotropic +Heisenberg benchmark at $T=1$ and $\eps=10^{-6}$. Locality suggests that many +symbolic densities extend to larger tori beyond the support radius, but the +current manifest does not claim a theorem uniform in $L$, anisotropy, time, or +tolerance. A scaling paper would require parameterized wraparound proofs, +analytic dependence of every ledger term, and certificates across a grid of +$(L,T,\eps)$ values. + +\subsection{Fixed product formula and cost model} + +The candidate and control use the same five-copy fourth-order Suzuki formula. +No search over modern optimized splittings is included. Similarly, the +three-CNOT bond cost is an upper-bound compilation model, not a full +fault-tolerant resource estimate. Native two-qubit gates, routing, parallel +depth, synthesis precision, error correction, and logical-state preparation +can change the operational optimum. Future work should make the cost model a +pluggable certificate field and report Pareto fronts for group count, two- +qubit gates, and depth. + +\subsection{Certificate size and discovery cost} + +The exact D4 and D5 ledgers are large. They are appropriate for an offline +proof artifact but not yet an interactive compiler pass. The deep verifier is +much simpler than the generator but still scales with every listed term and +pair checked within a group. Merkleized chunks, streaming verification, +symmetry-orbit proofs, and formally verified local rewrite kernels could +reduce storage and trust without replacing explicit evidence by an opaque +claim. + +\subsection{The fivefold boundary is open} + +A fivefold resource ratio would be obtained at $r=78$ because +\begin{equation} + \frac{11{,}791}{30\cdot78+1} + =\frac{11{,}791}{2{,}341} + =5.036736437419906\ldots. +\end{equation} +This is arithmetic, not a 78-step error certificate. The checked artifacts +identify three unresolved proof gates: a tighter translation-coupled D4 norm +at the 78-step budget, explicit D8 information with a delayed high-degree +tail, and a complete exact global ledger at $r=78$. Exact D5 grouping alone +does not discharge these obligations. + +\begin{figure}[t] + \centering + \includegraphics[width=0.8\textwidth]{figures/fivefold_gap.pdf} + \caption{Conditional error ledger at the fivefold resource threshold. The + stacked value exceeds the target; $r=78$ is therefore not certified. The + 97-step result remains the lowest accepted point of the current bound.} + \label{fig:fivefold-gap} +\end{figure} + +A disciplined optimization sequence is therefore: (1) profile the current +97-step error ledger; (2) improve the dominant D4 norm while preserving a +verifiable partition; (3) emit D8 explicitly; (4) move the geometric tail to a +higher starting degree; (5) regenerate exact contributions at 78; and (6) +accept the point only if the rational sum is at most $10^{-6}$. GPU or HPC +parallelism can accelerate term generation and graph search, but cannot +replace the last verifier step. + +\subsection{Optimality is relative to the bound} + +The rejection at 96 establishes adjacent minimality only under the emitted +upper bound. It is not a lower bound on the true operator error. Direct +large-system matrix simulation is infeasible, and lower-bound techniques have +not been specialized here to close the gap. Consequently the candidate may +still be physically accurate with fewer steps. The scientific claim concerns +what can be guaranteed, not an assertion that 97 executions are necessary. + +\subsection{Toward broader validation} + +The most valuable next experiments are not additional decimal precision at +the current point. They are adversarial reproductions: an independent Pauli +engine, certificates for open and periodic boundaries, anisotropic XXZ +couplings, alternate fragment orderings, and comparisons against optimized +splittings. These tests would reveal whether the dominant gains arise from +universal compiler structure or accidental symmetries of the isotropic +benchmark. diff --git a/tracks/qcs/solutions/WangTheoPhys/issue128/docs/manuscript/sections/norm_certificate.tex b/tracks/qcs/solutions/WangTheoPhys/issue128/docs/manuscript/sections/norm_certificate.tex new file mode 100644 index 000000000..d5f36e617 --- /dev/null +++ b/tracks/qcs/solutions/WangTheoPhys/issue128/docs/manuscript/sections/norm_certificate.tex @@ -0,0 +1,97 @@ +\section{Anticommuting Norm Certificates} +\label{sec:norm} + +After exact coefficient aggregation, the direct Pauli triangle inequality +$\opnorm{\sum_j c_jP_j}\leq\sum_j|c_j|$ remains sound but unnecessarily loose. +The leading right-generator defect contains many mutually anticommuting +strings. Their norm is an $\ell_2$, not an $\ell_1$, quantity. + +\subsection{Weighted anticommuting groups} + +\begin{lemma}[Pairwise-anticommuting Pauli norm] +\label{lem:anticommuting} +Let $P_1,\ldots,P_k$ be Hermitian Pauli operators with $P_i^2=I$ and +$\{P_i,P_j\}=0$ for $i\neq j$. For real coefficients $c_i$, +\begin{equation} + \opnorm{\sum_{i=1}^{k}c_iP_i}=\sqrt{\sum_{i=1}^{k}c_i^2}. + \label{eq:anticommuting-norm} +\end{equation} +\end{lemma} + +\begin{proof} +Squaring the sum gives +$\sum_i c_i^2I+\sum_{i\eps,\label{eq:adjacent}\ + R_\star&=\mathcal C(S,r_\star), +\end{align} +using exact integers, rational numbers, or outward rational intervals. The +second inequality does not prove that the physical circuit with +$r_\star-1$ fails; it proves that $r_\star$ is the smallest integer accepted +by this particular certified upper-bound function. This distinction is +retained throughout. diff --git a/tracks/qcs/solutions/WangTheoPhys/issue128/docs/manuscript/sections/related_work.tex b/tracks/qcs/solutions/WangTheoPhys/issue128/docs/manuscript/sections/related_work.tex new file mode 100644 index 000000000..077555003 --- /dev/null +++ b/tracks/qcs/solutions/WangTheoPhys/issue128/docs/manuscript/sections/related_work.tex @@ -0,0 +1,141 @@ +\section{Related Work and Positioning} +\label{sec:related} + +The relevant literature has diversified beyond a single sequence of tighter +versions of the same bound. Some works strengthen worst-case product-formula +analysis; others restrict the state or error metric; still others change the +simulation algorithm. A meaningful position therefore requires a taxonomy +rather than a one-dimensional ``SOTA'' label. + +\subsection{Direct product-formula bounds} + +Nearly optimal lattice simulation by high-order product formulas established +the favorable scaling of local formulas for geometrically local Hamiltonians +\citep{childs2019lattice}. The commutator-scaling framework of Childs et al. +then provided general explicit error representations in terms of nested +commutators \citep{childs2021trotter}. Model-specific evaluation has also +been developed for the Fermi--Hubbard model \citep{schubert2023hubbard}, while +strong-error analyses clarify the distinction between vector and operator +convergence for unbounded or chemistry-motivated settings +\citep{burgarth2024strong}. Lower-bound work studies when cancellation cannot +make the Trotter error arbitrarily smaller than upper estimates +\citep{hahn2024lower}. + +Our work remains in this direct, deterministic, operator-norm branch. It does +not introduce a new asymptotic theorem. Its contribution is an automated +finite-instance sharpening layer after the general theorem has identified the +relevant commutator structure. The proof object retains concrete +coefficients and Pauli relations that a broad theorem cannot economically +tabulate. + +\subsection{Restricted-state and average-case guarantees} + +Low-energy-subspace simulation can replace global energy scales by effective +low-energy quantities when the initial state is promised to lie in an +appropriate sector \citep{sahinoglu2021lowenergy}. Average-case analysis can +also yield speedups under distributional or norm-averaged criteria +\citep{chen2024average}. These guarantees may be more relevant for a given +application than a worst-case operator norm, but they answer a different +question. The present certificate makes no state promise and therefore +remains uniform over all input states. + +\subsection{Changing the approximation family} + +Multi-product formulas combine several product-formula circuits to cancel +leading errors. Recent work supplies explicit commutator scaling, dynamic +coefficient selection, and time-dependent extensions +\citep{zhuk2024dynamic,aftab2024mpf,mizuta2026mpf,mizuta2026timedependent}. +Chebyshev interpolation and randomized order-doubling similarly alter how +approximants are combined \citep{rendon2024chebyshev,cho2024doubling}. +Time-dependent product-formula constructions such as THRIFT exploit an +interaction-picture decomposition \citep{bosse2025thrift}. LCU-based Trotter +compensation and error-mitigation approaches add coherent or classical +correction layers \citep{zeng2025lcu,watson2025mitigation}; more recent nested- +commutator compensation seeks polylogarithmic precision without ancillas +\citep{wang2026hncc}. + +These methods can asymptotically or practically outperform repetition of a +single deterministic Suzuki step. Their costs include extra circuit +variants, sampling, negative or complex weights, success amplification, +ancillas, or different depth/gate accounting. Our certified result keeps the +original formula family fixed and reduces only the justified repetition +count. It is therefore complementary: the same proof-carrying norm compiler +could in principle supply coefficient bounds to an extrapolation or +compensation method, but such an extension is not established here. + +\subsection{Formula and ordering optimization} + +Optimized Trotter--Suzuki coefficient sets can lower the stage count or error +constant relative to recursive constructions \citep{malezic2026schemes}. +Commutation-aware ordering studies show that the placement of Heisenberg-type +terms can materially change realized error \citep{tate2026ordering}. +Practical error estimators trade some formal generality for computable +instance-level predictions \citep{maxwell2026practical}. Those directions +optimize the source formula or estimate its behavior. Here the source stage +sequence is fixed and the novelty is in rigorously compiling its complete +defect into a checkable upper bound. + +\begin{table}[p] +\centering +\small +\caption{Methodological comparison. ``Same target'' means a deterministic, +uniform operator-norm certificate for the fixed formula and compilation model +of \cref{sec:problem}.} +\label{tab:literature-map} +\begin{tabularx}{\textwidth}{@{}p{0.18\textwidth}p{0.18\textwidth}p{0.19\textwidth}p{0.18\textwidth}X@{}} +\toprule +Line of work & Main mechanism & Guarantee/cost change & Relation to our method & +Same target? \\ +\midrule +Commutator scaling \citep{childs2021trotter} & General nested-commutator error +representation & Worst-case direct product formula & Supplies the pinned +control and theoretical foundation & Yes, at theorem level \\ +Model-specific Hubbard bounds \citep{schubert2023hubbard} & Evaluate locality +and commutators for another model & Model-aware direct bound & Similar +model-specific philosophy; different Hamiltonian and certificate machinery & +Partly \\ +Low-energy simulation \citep{sahinoglu2021lowenergy} & Restrict input energy +sector & State-dependent promise & Could be combined with concrete algebra; +not used here & No \\ +Average-case speedup \citep{chen2024average} & Weaken worst-case metric to an +average & Distribution-dependent error & Orthogonal norm objective & No \\ +Dynamic/explicit MPF \citep{zhuk2024dynamic,aftab2024mpf,mizuta2026mpf} & +Linear combination of several formula lengths & New coefficients, sampling or +coherent combination & Our defect bounds may inform coefficients; current +circuit family differs & No \\ +Interpolation / randomization \citep{rendon2024chebyshev,cho2024doubling} & +Combine runs or random orderings & Changed estimator/algorithm & Potential +outer layer around certified formulas & No \\ +THRIFT \citep{bosse2025thrift} & Time-dependent interaction-picture formula & +Changed decomposition and circuit & Could benefit from proof-carrying +instance analysis & No \\ +LCU/mitigation \citep{zeng2025lcu,watson2025mitigation,wang2026hncc} & Correct +or cancel Trotter error & Ancilla, sampling, or compensation overhead & Uses +error structure actively; ours only reduces safe repetitions & No \\ +Formula/ordering optimization \citep{malezic2026schemes,tate2026ordering} & +Change stages, coefficients, or ordering & Different source circuit & +Complementary front end to our compiler & No \\ +This work & Complete-formula Lie extraction, exact Pauli aggregation, verified +anticommuting norms & Same formula; smaller certified repetition count & +Finite-instance proof-carrying sharpening & Yes \\ +\bottomrule +\end{tabularx} +\end{table} + +\subsection{Position of the present result} + +Within the narrow target class, this work establishes a strong finite-instance +result: a 4.050498$\times$ certified reduction over a pinned application of a +published general theorem, while preserving the formula, operator-norm metric, +and gate-accounting rule. Its scientific role is best described as a +\emph{model-aware proof compiler} between general theory and executable +resource estimation. + +It would be inaccurate to call the result an unconditional global state of the +art in Hamiltonian simulation. More recent algorithms can have superior +precision scaling or lower costs under changed assumptions. Conversely, +their headline numbers do not invalidate this result because they are not +necessarily certificates for the same fixed deterministic circuit. The +paper's defensible novelty is the combination of full-formula symbolic +cancellation, concrete Pauli norm compression, nonasymptotic closure, and an +independently checkable artifact. diff --git a/tracks/qcs/solutions/WangTheoPhys/issue128/docs/manuscript/sections/reproducibility.tex b/tracks/qcs/solutions/WangTheoPhys/issue128/docs/manuscript/sections/reproducibility.tex new file mode 100644 index 000000000..eacd7b3a4 --- /dev/null +++ b/tracks/qcs/solutions/WangTheoPhys/issue128/docs/manuscript/sections/reproducibility.tex @@ -0,0 +1,107 @@ +\section{Reproducibility and Artifact Protocol} +\label{sec:reproducibility} + +The repository is organized so that a reader can first validate the headline +claim in seconds and then choose progressively more expensive checks. No +private cluster is required to verify the submitted certificate. + +\subsection{Environment and commands} + +The reference implementation requires Python 3.11 or later and declares +NumPy, SciPy, and SymPy dependencies. From the \code{issue128} directory, the +standard sequence is: +\begin{verbatim} +python -m pip install -e '.[test]' +pytest -q +PYTHONPATH=src python scripts/verify.py \ + certificates/issue128-certificate.json +PYTHONPATH=src python scripts/build_d5_certificate.py --verify-only +PYTHONPATH=src python scripts/package_delivery.py --check +shasum -a 256 -c artifacts/SHA256SUMS +\end{verbatim} +The expensive algebraic replay is requested explicitly: +\begin{verbatim} +PYTHONPATH=src python scripts/verify.py \ + certificates/issue128-certificate.json --deep +\end{verbatim} +The main certificate can be regenerated with +\begin{verbatim} +PYTHONPATH=src python scripts/build_v3_certificate.py +\end{verbatim} +Long research-reproduction tests are marked \code{slow} and excluded by the +default Pytest configuration. This separation keeps ordinary continuous +integration responsive without disguising the cost of complete regeneration. + +\subsection{Claim-to-evidence map} + +\begin{longtable}{@{}p{0.27\textwidth}p{0.36\textwidth}p{0.30\textwidth}@{}} +\caption{Where each principal claim is established and how it is checked.} +\label{tab:claim-map}\\ +\toprule +Claim & Primary evidence & Independent check \\ +\midrule +\endfirsthead +\toprule +Claim & Primary evidence & Independent check \\ +\midrule +\endhead +393-step pinned control & Baseline rational and center in +\code{issue128-certificate.json} & Deep regeneration of all 31 centers \\ +97 accepted, 96 rejected & Exact global and previous-step rational pairs in +the main certificate & Exact comparison against $1/10^6$ \\ +75,324 D4 terms and 7,576 groups & Hashed +\code{issue128-d4-groups.json} & Coverage, symplectic pair checks, and rational +square-root replay \\ +D4 bound 6.472926505087888 & Group membership and coefficient endpoints & +Verifier squares each upper root and resums \\ +11,791 and 2,911 groups & Step counts plus $G(r)=30r+1$ & Direct integer +identity checks \\ +Bond and CNOT totals & Cost-model fields and claimed-resource block & +$72G(r)$ and $216G(r)$ recomputation \\ +4.050498$\times$ improvement & Exact pair $(11791,2911)$ & Fraction comparison, +not rounded decimal \\ +Small-system consistency & Dense $2\times2$ artifact & Independent matrix +exponential and operator norm \\ +D5 follow-up only & Hashed compressed D5 sidecar and summary status & +\code{--verify-only}; explicit \code{not\_certified} fivefold status \\ +\bottomrule +\end{longtable} + +\subsection{Immutable bindings} + +At the frozen delivery point the primary digests are +\begin{align*} +\mathrm{SHA256}(\text{main certificate}) + &=\code{0a09623ce3b292a3637065c870fb3153b}\ldots,\\ +\mathrm{SHA256}(\text{D4 sidecar}) + &=\code{a397414bb0229fb1ebdb38798aa781fb8}\ldots,\\ +\mathrm{SHA256}(\text{D5 sidecar}) + &=\code{c5e8968a93b4497b41fe42c0e364324}\ldots. +\end{align*} +The complete 64-hex-character values are stored in the package manifest and +summary. A printed prefix is for human orientation only; verification uses +the complete digest. + +The manuscript is a derived view and is not part of the frozen proof bundle. +If the manuscript, figures, or wording changes, the mathematical claim remains +bound to the same JSON and sidecars. Conversely, any change to a certificate +requires regeneration of its digest and all summaries that cite it. + +\subsection{Machine-readable and human-readable outputs} + +The delivery contains a main certificate JSON, large exact sidecars, a compact +JSON summary, a plain-text summary, verification transcript, checksums, source +code, tests, a short technical report, and this long-form manuscript. The +machine-readable objects are authoritative for exact numerators and +denominators. Human documents explain the method and use outward decimal +summaries; they should never be parsed as the proof source. + +\subsection{Research integrity statement} + +Software-assisted symbolic algebra, exact arithmetic, automated tests, and +language-model-assisted drafting were used in preparing the artifact and +manuscript. Scientific responsibility remains with the author. Every +headline numerical statement is checked against the frozen certificate by a +separate manuscript validation script; literature statements are cited to +their primary publications. No generative system output is accepted as a +mathematical witness without executable or deductive verification. diff --git a/tracks/qcs/solutions/WangTheoPhys/issue128/docs/manuscript/sections/results.tex b/tracks/qcs/solutions/WangTheoPhys/issue128/docs/manuscript/sections/results.tex new file mode 100644 index 000000000..813bc02f4 --- /dev/null +++ b/tracks/qcs/solutions/WangTheoPhys/issue128/docs/manuscript/sections/results.tex @@ -0,0 +1,156 @@ +\section{Certified Results} +\label{sec:results} + +This section reports only values bound to the checked-in certificate. The +primary metric is the compiled CNOT upper bound; merged fragment +exponentials and bond propagators are reported because they make the integer +cost model transparent. + +\subsection{Headline resource reduction} + +The candidate certificate accepts $r=97$ steps. Adjacent identical fragment +exponentials merge across recursive boundaries, so a 31-stage step sequence +contains +\begin{equation} + G(r)=30r+1 + \label{eq:merged-groups} +\end{equation} +merged group exponentials. Each group is one perfect matching with 72 bonds. +The selected two-qubit bond-propagator compilation is charged at three CNOTs. +Therefore +\begin{equation} + B(r)=72G(r),\qquad C(r)=3B(r)=216G(r). + \label{eq:resource-map} +\end{equation} + +\begin{table}[t] +\centering +\caption{Certified resource comparison under the common compilation model.} +\label{tab:headline-results} +\begin{tabular}{@{}lrrrr@{}} +\toprule +Method & Steps & Merged groups & Bond propagators & CNOT upper \\ +\midrule +Pinned published-theorem control & 393 & 11,791 & 848,952 & 2,546,856 \\ +Proof-carrying model-aware bound & 97 & 2,911 & 209,592 & 628,776 \\ +\midrule +Candidate/control & 0.2468 & 0.2469 & 0.2469 & 0.2469 \\ +\bottomrule +\end{tabular} +\end{table} + +Because the last three resource measures are proportional under +\cref{eq:resource-map}, their exact improvement factor is +\begin{equation} + \frac{11{,}791}{2{,}911}=4.050498110614909\ldots. + \label{eq:exact-ratio} +\end{equation} +Equivalently, 8,880 of 11,791 merged groups are removed, a reduction of +75.31167839877873\%. The claimed multiplier is an exact rational comparison; +the decimal is for readability. + +\begin{figure}[t] + \centering + \includegraphics[width=0.94\textwidth]{figures/resources.pdf} + \caption{Resources for the pinned theorem instantiation and the certified + model-aware result. Values are exact integers; axes are scaled separately + to keep the three compiled metrics readable.} + \label{fig:resources} +\end{figure} + +\subsection{Finite-step error ledger} + +At 97 steps the global upper bound is the sum of five outward rational +contributions shown in \cref{tab:error-ledger}. The leading D4 term is only +about half the final value; D5--D7 and the tail are quantitatively material. +This demonstrates why a leading-order-only prediction would not be an +adequate certificate at the boundary. + +\begin{table}[t] +\centering +\caption{Outward decimal summaries of exact rational contributions at +$r=97$.} +\label{tab:error-ledger} +\begin{tabular}{@{}lrr@{}} +\toprule +Contribution & Global error upper & Fraction of total (approx.) \\ +\midrule +D4 & $5.264367936822258\times10^{-7}$ & 52.86\% \\ +D5 & $1.383487536744612\times10^{-7}$ & 13.89\% \\ +D6 & $1.579057000594836\times10^{-7}$ & 15.86\% \\ +D7 & $1.894657080691470\times10^{-8}$ & 1.90\% \\ +Tail & $1.542560312083473\times10^{-7}$ & 15.49\% \\ +\midrule +Total & $9.958938494314325\times10^{-7}$ & 100\% \\ +\bottomrule +\end{tabular} +\end{table} + +\begin{figure}[t] + \centering + \includegraphics[width=0.72\textwidth]{figures/error_ledger.pdf} + \caption{Finite-step error ledger at the accepted integer. The dashed line + marks the total tolerance, not an independent allowance for each term.} + \label{fig:error-ledger} +\end{figure} + +The adjacent-step evidence is decisive for the integer search: +97 is accepted at $9.958938494314325\times10^{-7}$, whereas the same bound at +96 evaluates to $1.050565873970784\times10^{-6}$. The margin at 97 is about +$4.11\times10^{-9}$, so outward arithmetic and tail closure are not cosmetic; +ordinary double-precision summaries are not the source of the comparison. + +\subsection{Algebraic compression statistics} + +The D4 group sidecar contains 75,324 coefficient-bearing Pauli terms divided +into 7,576 verified groups. The largest group has ten members. The grouped +translation-cell norm $6.472926505087888$ replaces a post-aggregation direct +Pauli-$\ell_1$ value $20.160968407335066$, giving the local factor +$3.114660484942633$. This local factor and the final resource factor need not +coincide: the latter also reflects the nonlinear inversion of the step bound, +the subleading terms, the tail, and the integer merge rule. + +An auxiliary exact D5 computation produces 605,832 terms, 123,106 groups, and +a per-site upper bound $11.23706750025697$. It is retained as evidence for a +future tightening effort. It does not replace the complete D4--D7 global +ledger and is not used to claim a lower accepted step count. + +\subsection{Small-system structural cross-check} + +On a degenerate periodic $2\times2$ instance at the same 97-step count, direct +dense evaluation gives operator-norm error +$9.128277711816321\times10^{-10}$. The schema-v3 certificate specialization +gives the upper bound $2.7666666666666666\times10^{-8}$. The latter dominates +the measured error and remains below $10^{-6}$. This calculation checks stage +order, normalization, periodic bookkeeping, and the direction of the +inequality using an independent dense route. It is not an empirical proof of +the $12\times12$ result. + +\subsection{Recursive Suzuki audit} + +For context, the main certificate records a separate audit of standard +recursive Suzuki orders under its declared audit assumptions. The values in +\cref{tab:recursive-audit} are taken from the manifest-bound schema-v3 JSON; +they are not used in the primary 4.05$\times$ comparison. + +\begin{table}[t] +\centering +\caption{Auxiliary recursive-order audit stored in the authoritative main +certificate.} +\label{tab:recursive-audit} +\begin{tabular}{@{}rrrrr@{}} +\toprule +Order & Steps & Stages/step & Merged groups & Error upper summary \\ +\midrule +2 & 29,964 & 7 & 179,785 & $9.9999203748666826\times10^{-7}$ \\ +4 & 808 & 31 & 24,241 & $9.9682621337975879\times10^{-7}$ \\ +6 & 370 & 151 & 55,501 & $9.8463753882538689\times10^{-7}$ \\ +8 & 335 & 751 & 251,251 & $9.9829085642259163\times10^{-7}$ \\ +\bottomrule +\end{tabular} +\end{table} + +This audit illustrates that increasing formal order does not monotonically +reduce compiled cost at a fixed precision: the recursive stage multiplier can +outgrow the reduction in steps. It should not be confused with a literature- +wide optimal-formula search. diff --git a/tracks/qcs/solutions/WangTheoPhys/issue128/docs/manuscript/sections/verification.tex b/tracks/qcs/solutions/WangTheoPhys/issue128/docs/manuscript/sections/verification.tex new file mode 100644 index 000000000..f02e6bc76 --- /dev/null +++ b/tracks/qcs/solutions/WangTheoPhys/issue128/docs/manuscript/sections/verification.tex @@ -0,0 +1,108 @@ +\section{Certificate Architecture and Independent Verification} +\label{sec:verification} + +Large symbolic calculations are difficult to trust by inspection. We use a +proof-carrying architecture: discovery may be complex, but the final claim is +accepted only after a narrower verifier rechecks explicit evidence. + +\subsection{Trusted and untrusted components} + +\begin{figure}[t] +\centering +\resizebox{\textwidth}{!}{% +\begin{tikzpicture}[ + node distance=6mm and 10mm, + box/.style={draw,rounded corners,align=center,minimum height=9mm,minimum width=33mm}, + untrusted/.style={box,fill=orange!10}, + trusted/.style={box,fill=green!10}, + arr/.style={-{Stealth[length=2mm]},thick} +] +\node[untrusted] (gen) {symbolic generator\\and grouping search}; +\node[untrusted,below=of gen] (hpc) {parallel/HPC discovery\\and cached ledgers}; +\node[box,right=of gen,yshift=-8mm,fill=blue!5] (bundle) {hashed certificate bundle\\JSON, groups, summaries}; +\node[trusted,right=of bundle] (fast) {fast verifier\\schema + arithmetic}; +\node[trusted,below=of fast] (deep) {deep replay\\algebra + group membership}; +\node[trusted,right=of fast,yshift=-8mm] (claim) {accepted claim\\or explicit failure}; +\draw[arr] (gen)--(bundle); +\draw[arr] (hpc)--(bundle); +\draw[arr] (bundle)--(fast); +\draw[arr] (bundle)--(deep); +\draw[arr] (fast)--(claim); +\draw[arr] (deep)--(claim); +\end{tikzpicture} +} +\caption{Trust boundary. Parallel discovery and heuristic grouping are +untrusted producers. The accepted result depends on explicit evidence and +two verification paths.} +\label{fig:trust-boundary} +\end{figure} + +The trusted claim comprises the mathematical lemmas in this paper, the small +verification programs, exact integer/rational arithmetic, and the cryptographic +binding between manifest and sidecars. It does not require trusting the job +scheduler, the heuristic's optimality, cached stdout, or a claimed successful +HPC run. + +\subsection{Certificate contents} + +The schema-v3 certificate records: +\begin{itemize} + \item the complete benchmark tuple, normalization, formula, time, tolerance, + and primary resource metric; + \item the pinned theorem instantiation used for the baseline; + \item coefficient precision policies and rational interval endpoints; + \item D4 group-certificate path, SHA-256 digest, term count, group count, + maximum size, and norm enclosure; + \item D4--D7 and tail contributions at the accepted integer; + \item the accepted and previous-step global-error rationals; + \item integer resource arithmetic and the exact improvement ratio; and + \item auxiliary audit results explicitly separated from the primary claim. +\end{itemize} +Large membership data are sidecars rather than embedded in prose. The +manifest hash prevents a reviewer from inadvertently validating a different +file with the same name. + +\subsection{Fast and deep modes} + +Fast verification checks the schema, paths, hashes, integer identities, +rational comparisons, resource formulas, and headline invariants. It is +intended for continuous integration and reviewer orientation. Deep +verification additionally replays coefficient aggregation, checks every group +for exact coverage and pairwise anticommutation, verifies all rational square +roots, and reproduces the finite-step sum. + +The two modes serve different purposes. Fast verification answers whether a +checked-in bundle is internally bound and arithmetically self-consistent. +Deep verification answers whether the large algebraic witness supports its +declared norm. Neither mode treats a rounded decimal in a PDF as evidence. + +\subsection{Negative tests and failure localization} + +A certificate system is useful only if nearby corruptions fail. The test +suite mutates representative fields: a group member, a coefficient endpoint, +the accepted step count, a resource total, and a sidecar digest. Each mutation +must cause a nonzero verifier result. Further unit tests cover Pauli phase +multiplication, symplectic commutation, periodic canonicalization, rational +square-root enclosure, and the local 16-Pauli lemma. + +Failures are reported by layer. A hash mismatch is distinguished from a +schema error; a duplicate group member from a non-anticommuting pair; a failed +tail precondition from an error-bound comparison. This makes independent +reproduction tractable: a reviewer need not rerun the entire discovery +pipeline to diagnose a damaged artifact. + +\subsection{Claim classes} + +We label statements by evidence class: +\begin{description} + \item[Certified.] Derived from a manifest-bound witness and accepted by the + verifier, such as the 97-step result and integer resource counts. + \item[Cross-checked.] Supported by a structurally independent small-system or + exhaustive local calculation, but not the main large-system proof. + \item[Conditional.] An arithmetic consequence under a stated future premise, + such as the norm budget required for a fivefold result. + \item[Prospective.] A research direction or engineering optimization with no + claimed current performance. +\end{description} +This vocabulary prevents an observed trend, a possible optimization, and a +completed theorem from being conflated in the abstract or conclusion. diff --git a/tracks/qcs/solutions/WangTheoPhys/issue128/docs/plans/2026-07-29-anticommuting-d4-certificate-design.md b/tracks/qcs/solutions/WangTheoPhys/issue128/docs/plans/2026-07-29-anticommuting-d4-certificate-design.md new file mode 100644 index 000000000..ed93ba96c --- /dev/null +++ b/tracks/qcs/solutions/WangTheoPhys/issue128/docs/plans/2026-07-29-anticommuting-d4-certificate-design.md @@ -0,0 +1,164 @@ +# Anticommuting D4 Certificate Design + +## Objective + +Tighten the existing schema-v3 certificate for the fixed benchmark + +```text +periodic 12x12 spin-1/2 isotropic Heisenberg model +h_ij = (XX+YY+ZZ)/4 +T = 1 +operator-norm tolerance = 1e-6 +``` + +without changing its five-copy fourth-order Suzuki circuit, four-matching +fragmentation, or compiled-cost convention. The immediate target is a +machine-verified candidate at 98 Trotter steps: + +```text +candidate groups = 30*98 + 1 = 2941 +published groups = 11791 +improvement = 11791/2941 = 4.00918055083305... +``` + +## Evidence and feasibility + +At 98 steps the existing certificate gives a degree-four contribution +`1.6100509989613002e-6`. A discovery-only greedy partition of the concrete +local degree-four defect into pairwise-anticommuting groups of at most ten +Pauli strings reduced this contribution by a factor +`3.108054463616103`. Leaving every other certified contribution unchanged +then gives the discovery estimate + +```text +degree-four contribution: 5.180253492366627e-7 +all unchanged contributions: 4.395310649104868e-7 +total: 9.575564141471495e-7 +``` + +The `4.2443585852850405e-8` margin is much larger than the existing rational +coefficient-interval widths. This discovery result is not itself a proof; +the design below converts it into an independently checked certificate. + +## Mathematical certificate + +Write the exact local right-generator degree-four coefficient as + +```text +D4 = sum_j c_j P_j, +``` + +where every `P_j` is a phase-free Pauli string and every `c_j` is enclosed by +a rational interval. Partition the indices into disjoint groups `G_a` such +that every pair of distinct Paulis in one group anticommutes. Since + +```text +(sum_{j in G_a} c_j P_j)^2 = (sum_{j in G_a} c_j^2) I, +``` + +the group has the rigorous bound + +```text +||sum_{j in G_a} c_j P_j|| + <= sqrt(sum_{j in G_a} sup(|c_j|)^2). +``` + +One triangle inequality between groups gives the local certificate + +```text +||D4|| <= sum_a sqrt(sum_{j in G_a} sup(|c_j|)^2). +``` + +Every square root is rounded upward to a rational number. The optimizer may +use floating point to discover the partition, but the verifier trusts only +the submitted Pauli identifiers, exact symplectic anticommutation checks, +rational coefficient intervals, and outward-rounded square-root witnesses. + +## Architecture + +### Exact coefficient construction + +Refactor the existing leading-E5 calculation so it can return its merged +Pauli coefficient intervals instead of immediately summing their absolute +values. Canonicalize every local Pauli under translations by the colored +`2x2` unit cell before merging. Multiplying the homogeneous log coefficient +by five produces the degree-four right-generator coefficient. + +The original Pauli-l1 result remains available and remains the bound used for +the degree-five, degree-six, and degree-seven repeated-adjoint estimates. +Only `d4_site` changes in the first implementation. + +### Partition discovery + +Sort Pauli terms by decreasing interval midpoint magnitude. Starting with +the largest unused term, greedily add the largest unused term that +anticommutes with every current group member until the configured maximum +group size is reached. Singleton groups are allowed. Discovery output is a +deterministic tuple of Pauli-index tuples. + +### Proof object + +The schema-v3 candidate gains: + +```text +d4_norm_method +d4_pauli_terms +d4_anticommuting_groups +d4_group_bounds +d4_cell_norm_upper +``` + +Pauli terms are stored in binary symplectic form with rational coefficient +interval endpoints. Group bounds are exact rational outward upper bounds. + +### Independent verification + +The fast verifier checks: + +1. every submitted Pauli term appears exactly once; +2. indices are in range and groups are nonempty; +3. every pair within a group anticommutes exactly; +4. every group bound squares to at least the sum of squared coefficient + absolute upper endpoints; +5. group bounds sum to the submitted cell bound; +6. the cell-to-site conversion and total-error arithmetic are exact; +7. step 98 meets the tolerance and step 97 does not. + +Deep verification regenerates the interval coefficient map and requires it +to equal the proof object before repeating all fast checks. + +## Failure containment + +The new certificate is accepted only if its exact total at 98 steps is at +most `1e-6`. If interval inflation removes the discovery margin: + +1. increase the maximum anticommuting group size; +2. improve the deterministic clique assignment while retaining exact + verification; then +3. apply the same certificate to degree five. + +At every point the prior 116-step schema-v3 certificate remains a valid +fallback. No empirical norm is used in the proof path. + +## Tests + +- Exact unit tests for symplectic anticommutation and outward square roots. +- Partition coverage and corruption-rejection tests. +- A regression test proving the grouped bound is no larger than Pauli-l1. +- A slow regeneration test for the exact D4 interval coefficient map. +- Fast and deep certificate-verifier tests. +- Integer-boundary tests at steps 98 and 97. +- Existing dense `2x2` evolution cross-check updated to step 98. +- Full solution and repository test suites before submission. + +## Deferred work + +The 5x and 10x programs are intentionally separate: + +- 5x: exact right-generator coefficients through degree 10 plus shorter tail + and finite-cluster norm certificates; +- 10x: matching-phase cycling and a degree-four processor, or an + interval-certified optimized sixth-order kernel. + +Those changes are not required to establish the 4x certificate and will not +be mixed into its first reviewable commit series. diff --git a/tracks/qcs/solutions/WangTheoPhys/issue128/docs/plans/2026-07-31-proof-carrying-trotter-paper-design.md b/tracks/qcs/solutions/WangTheoPhys/issue128/docs/plans/2026-07-31-proof-carrying-trotter-paper-design.md new file mode 100644 index 000000000..e761efef2 --- /dev/null +++ b/tracks/qcs/solutions/WangTheoPhys/issue128/docs/plans/2026-07-31-proof-carrying-trotter-paper-design.md @@ -0,0 +1,205 @@ +# Proof-Carrying Trotter Bound Paper Design + +## Objective + +Create an independent 22--30 page English research manuscript that develops +the Issue 128 result into a general scientific narrative. The manuscript will +present a proof-carrying, model-aware compiler for rigorous Trotter resource +bounds and use the certified `393 -> 97` Heisenberg result as its principal +case study. It will not modify the frozen certificate or promote the +conditional fivefold arithmetic to a certified claim. + +## Audience and publication posture + +The primary audience is researchers in quantum algorithms, Hamiltonian +simulation, computational many-body physics, and computer-assisted proof. +The initial format is a self-contained generic LaTeX article suitable for an +arXiv `quant-ph` release. Its structure will support later adaptation to +Physical Review Research or Quantum without rewriting the scientific core. + +The paper will make a strong but scoped claim: for the fixed periodic +`L=12` square-lattice Heisenberg benchmark, fixed five-copy fourth-order +Suzuki formula, uniform operator-norm target, and fixed compilation model, the +submitted exact certificate reduces the accepted resource count from 11,791 +to 2,911 merged group exponentials. The paper will call this a best-known +certified bound only with an explicit "to the best of our knowledge" qualifier. + +## Scientific narrative + +The paper is organized around a compiler pipeline rather than a chronological +competition report. Its source language is a local Pauli Hamiltonian, +fragmentation, product formula, evolution time, tolerance, and cost model. +Its intermediate representations are free associative words, homogeneous Lie +defects, concrete symplectic Pauli ledgers, and rational norm certificates. +Its output is an independently checkable adjacent-step error certificate and +an integer resource count. + +Every optimization stage will use the same explanatory pattern: + +1. identify the conventional relaxation; +2. explain what structural information it discards; +3. define the replacement transformation; +4. state and justify its soundness; +5. describe the implementation and data representation; +6. identify what the independent verifier checks; and +7. quantify the contribution to the final result where an isolated number is + available. + +## Claim taxonomy + +All statements will be labeled conceptually as one of three classes: + +- **Certified:** exact rational or outward interval statements verified from + frozen artifacts, including the `r=97` pass, `r=96` rejection, resource + arithmetic, D4 partition, D5 sidecar identity, and small-system enclosure. +- **Empirical cross-check:** dense small-system numerical error and any timing + or implementation measurements that are not used in the theorem. +- **Conditional research direction:** the `r=78` fivefold arithmetic, + processor candidates, shorter product formulas, and delayed-tail proposals. + +No conditional item may appear in the abstract or conclusion as an achieved +resource improvement. + +## Planned manuscript structure + +1. **Abstract.** State the fixed benchmark, proof-compiler contribution, + `393 -> 97` boundary, exact ratio, and trust model. +2. **Introduction.** Motivate the gap between general commutator bounds and + finite-instance resource estimates; list contributions and limitations. +3. **Problem formulation.** Define the Hamiltonian, fragmentation, product + formula, operator-norm objective, and compilation cost. +4. **Baseline reconstruction.** Separate the published Childs--Su--Tran-- + Wiebe--Zhu theorem from the paper's pinned interval instantiation; explain + the 31-center scan and the `r=393` control. +5. **Compiler overview.** Present inputs, intermediate representations, + outputs, invariants, and the discovery/trust boundary. +6. **Exact product-formula algebra.** Define sparse free-word series, + truncated multiplication, formal logarithms, symmetry, and order checks. +7. **Lie projection.** Define the Dynkin--Specht--Wever projection and its use + for degree-five and degree-seven defects. +8. **Concrete Pauli evaluation.** Define symplectic Pauli multiplication, + local commutator expansion, support filtering, and translation cells. +9. **Norm-last optimizations.** Explain translation canonicalization, exact + coefficient aggregation, and cancellation before norm inequalities. +10. **Certified anticommuting partitions.** Define the graph, clique cover, + Euclidean group norm, exact coverage checks, and rational square-root + enclosure; report the D4 statistics and reduction. +11. **Heisenberg local lemma.** Prove the 16-case bond commutator result and + explain its cumulative use in higher-degree propagation. +12. **Finite-step certificate.** Derive the right logarithmic generator, + degrees D4--D7, Duhamel integration, support growth, and rational + geometric tail. +13. **Resource compilation.** Prove `G(r)=30r+1`, derive bond and CNOT counts, + and establish the adjacent integer boundary. +14. **Proof-carrying verification.** Specify certificate schemas, hashes, + independent verifier checks, negative tests, and trusted arithmetic. +15. **Results and ablations.** Present the error ledger, resource table, + direct-versus-grouped D4 norm, recursive-order audit, negative routes, + and small-system cross-check. +16. **Related work and positioning.** Distinguish worst-case commutator bounds, + model-specific symbolic bounds, low-energy/average-case analyses, + MPF/LCU/error mitigation, formula optimization, and ordering methods. +17. **Limitations and research frontier.** Quantify the D4/D5/D6/D7/tail + bottlenecks at `r=78` and explain why tenfold requires an order lift. +18. **Reproducibility.** Provide exact commands, artifact identities, platform + metadata, and a claim-to-evidence matrix. +19. **Conclusion.** Restate the scoped certified contribution. +20. **Appendices.** Include formal-series identities, verifier pseudocode, + certificate schema, local Pauli table, detailed ledgers, and additional + literature-comparison tables. + +## Figures and tables + +The paper will include at least the following visuals: + +- compiler pipeline from formula to certificate; +- trusted versus untrusted computation boundary; +- baseline and candidate resource bars; +- stacked D4--tail error ledger at `r=97`; +- D4 norm reduction from termwise l1 to certified grouped l2; +- fivefold bottleneck ledger at `r=78` with certified/non-certified labeling; +- method comparison matrix across operator norm, changed circuit, state + restrictions, and machine verification; +- claim-to-evidence table and main artifact hashes. + +Figures will be generated from frozen numerical values or from deterministic +source data embedded with explicit provenance. Decorative figures will not +be used. + +## File architecture + +The manuscript will live under `docs/manuscript/` and remain separate from +the competition report: + +```text +docs/manuscript/ + main.tex + references.bib + sections/ + abstract.tex + introduction.tex + problem.tex + baseline.tex + compiler.tex + free_lie.tex + pauli.tex + norm_certificate.tex + finite_step.tex + verification.tex + results.tex + related_work.tex + limitations.tex + reproducibility.tex + conclusion.tex + appendices.tex + figures/ + scripts/ + output/pdf/issue128-proof-carrying-trotter-paper.pdf +``` + +Each section file will have one clear responsibility. The manuscript will +reference, but not duplicate or rewrite, the frozen JSON certificates. + +## Evidence policy + +Numerical claims must be sourced from one of: + +- `certificates/issue128-certificate.json`; +- `certificates/issue128-d4-groups.json`; +- `certificates/issue128-d5-groups.json.gz`; +- `certificates/issue128-small-crosscheck.json`; +- `artifacts/issue128-summary.json`; +- `artifacts/verification-transcript.txt`; or +- the committed fivefold audit. + +Literature claims require primary-source citations. The bibliography will +cover the 2021 commutator-scaling baseline, model-specific bounds, +low-energy and average-case results, multi-product formulas, randomized +corrections, interpolation and error mitigation, optimized compositions, +ordering strategies, practical BCH estimation, and nested-commutator +compensation. + +## Verification and quality gates + +The manuscript is complete only when all of the following hold: + +1. LaTeX compiles without errors, unresolved references, or unresolved + citations. +2. The PDF has at least 15 substantive pages. +3. Every page is rendered to an image and visually checked for clipping, + overlap, broken equations, illegible plots, and bad page breaks. +4. Automated text checks find no `TODO`, placeholder citation, or accidental + fivefold certified claim. +5. All copied resource and error numbers match the frozen JSON artifacts. +6. The manifest-bound competition package still passes unchanged. +7. The final handoff identifies the manuscript as a local follow-up document, + not as a post-deadline mutation of the judged PR. + +## Non-goals + +- Do not alter the frozen `4.050498x` certificate or its manifest. +- Do not claim an accepted `r=78` certificate. +- Do not invent multi-model numerical results that have not been computed. +- Do not present the current method as an asymptotic improvement over all + Hamiltonian-simulation algorithms. +- Do not commit the unrelated in-progress HPC files as part of the paper. diff --git a/tracks/qcs/solutions/WangTheoPhys/issue128/docs/presentation/issue128-10-minute-talk.md b/tracks/qcs/solutions/WangTheoPhys/issue128/docs/presentation/issue128-10-minute-talk.md new file mode 100644 index 000000000..46cf2e0b6 --- /dev/null +++ b/tracks/qcs/solutions/WangTheoPhys/issue128/docs/presentation/issue128-10-minute-talk.md @@ -0,0 +1,129 @@ +# Issue 128：10 分钟成果展示讲稿 + +## 演讲边界 + +- 主张只冻结为：同一基准、同一四阶 Suzuki 公式、同一编译成本模型下，合并群指数从 11,791 降为 2,911，精确倍率为 `11791/2911 = 4.050498110614909...`。 +- `r=97` 被精确证书接受，`r=96` 被同一误差函数拒绝。 +- D8/HPC 计算属于“正在验证的五倍候选门槛”，在完整归并和全局误差账本通过前，不宣布五倍结果。 + +## 0:00--0:40｜第 1 页：一句话结论 + +屏幕只放四个数字：`393 → 97`、`11791 → 2911`、`4.050498×`、`ε = 10⁻⁶`。 + +讲稿： + +> 我解决的是一个固定的二维 Heisenberg Trotter 基准。没有换 Hamiltonian，没有换乘积公式，也没有换成本口径。公开高阶交换子界需要 393 步；我的机器可验证证书接受 97 步并拒绝 96 步。因此合并群指数从 11,791 降到 2,911，精确提升 4.050498 倍。 + +## 0:40--1:35｜第 2 页：题目与公平比较 + +放基准表：`L=12`、`N=144`、`T=1`、周期边界、各向同性 Heisenberg、全局算符范数误差 `≤10⁻⁶`。右侧放成本公式 `G(r)=30r+1`。 + +讲稿重点： + +> 四个匹配片段、31-stage 四阶 Suzuki 单步和每条 Heisenberg bond 的编译上界，两边完全相同。倍率来自证明更紧，而不是问题变简单。用 `G(r)=30r+1` 代入 393 和 97，就得到可复核的整数资源比。 + +## 1:35--2:30｜第 3 页：旧证明为什么松 + +画一条对比流水线： + +`先三角不等式 → 取消被破坏 → 大常数` + +对比 + +`完整公式对数 → Lie 投影 → 具体 Pauli 合并 → 最后取范数` + +讲稿： + +> 关键不是把同一个程序跑快，而是推迟范数不等式。若先把嵌套交换子逐项取绝对值，Heisenberg 代数中的符号取消和同 Pauli 字符串合并会永久丢失。我先保留精确有理系数，到具体格点表示中完成合并，再做可证明的范数闭合。 + +## 2:30--4:05｜第 4 页：核心技术 + +只讲三件事： + +1. 31-stage 公式的自由结合代数对数与 Lie 投影； +2. 平移规范化后合并完全相同的 Pauli 字符串； +3. 对两两反对易组使用 `||Σ c_j P_j|| = sqrt(Σ|c_j|²)`，平方根向外取有理上界。 + +讲稿： + +> D4 侧车中有 75,324 个规范 Pauli 项，被证书划分成 7,576 个两两反对易组，最大组大小 10。逐项 Pauli-l1 的局部上界是 20.1609，而分组后的认证上界是 6.4729，局部范数步骤收紧 3.11466 倍。分组算法可以是启发式的，但 verifier 会独立检查覆盖、重复、每一对反对易关系以及每个向外平方根界。 + +## 4:05--5:25｜第 5 页：严格误差账本 + +放 `r=97` 的五行表： + +| 项 | 全局上界 | +|---|---:| +| D4 | `5.2643679368e-7` | +| D5 | `1.3834875367e-7` | +| D6 | `1.5790570006e-7` | +| D7 | `1.8946570807e-8` | +| D8+ tail | `1.5425603121e-7` | +| 总计 | `9.9589384943e-7 < 1e-6` | + +讲稿： + +> 所有判定使用整数交叉相乘和精确有理数，屏幕上的小数只用于阅读。尤其重要的是，没有把七阶以上直接删掉：D8 及以上由显式、向外的有理几何尾闭合。97 步严格低于阈值，而 96 步是 `1.0505658739e-6`，严格高于阈值。 + +## 5:25--6:35｜第 6 页：结果与题目符合性 + +放资源表： + +| 资源 | 对照 | 本工作 | 倍率 | +|---|---:|---:|---:| +| Trotter steps | 393 | 97 | 4.0515×（步数口径） | +| merged groups | 11,791 | 2,911 | 4.050498× | +| bond propagators | 848,952 | 209,592 | 4.050498× | +| CNOT upper bound | 2,546,856 | 628,776 | 4.050498× | + +讲稿： + +> 题目要求至少两倍；这里达到四倍以上。主倍率必须报资源整数比 `11791/2911`，不能用四舍五入后的误差或只报步数比。所有成本行由同一编译模型导出，所以口径一致。 + +## 6:35--8:10｜第 7 页：现场可验证性 + +现场只运行快速 verifier，不现场重算大侧车： + +```bash +pytest -q +PYTHONPATH=src python scripts/verify.py \ + certificates/issue128-certificate.json +shasum -a 256 -c artifacts/SHA256SUMS +``` + +先展示预期字段：`valid=true`、`verification_level=fast`、 +`deep_proof_regenerated=false`、`published_steps=393`、 +`candidate_steps=97`、`exact_improvement_ratio=11791/2911`。 + +现场快速命令的 `deep_proof_regenerated` 预期为 `false`；预先录制或单独运行的 +`--deep` 结果才应为 `true`。十分钟演讲不等待 deep regeneration 完成。 + +讲稿： + +> 搜索代码不在信任边界里。verifier 从侧车重新检查哈希、精确覆盖、反对易关系、误差账本和资源整数算术。今天本地常规测试结果是 93 passed、11 deselected；深验证也重新生成并通过。大文件由 SHA-256 绑定，报告、JSON、文本摘要和执行 transcript 相互对应。 + +## 8:10--9:15｜第 8 页：正在推进的五倍门槛 + +放一条红线：`5× ⇔ r≤78`，并写 `尚未声明`。 + +讲稿： + +> 五倍的资源算术要求 78 步或更少，但算术不是误差证明。现有精确 D5 侧车已经通过；当前超算任务正在把完整右生成元的 D8 系数按 31 个 stage 精确分片计算，并用正反两种顺序归并检查确定性。只有 D4 到 D8+ 的全局账本在 78 步整体小于 `10⁻⁶`，我才会改写倍率；否则结果仍冻结为 4.050498 倍。 + +## 9:15--10:00｜收束 + +讲稿： + +> 这项工作的结论不是“数值上看起来更好”，而是“同一问题上，把 4.050498 倍资源下降变成独立程序能够拒绝错误、重新生成关键量并复核整数边界的证书”。现有四倍结果已经闭环；五倍是一个有明确通过条件、不会提前宣传的后续研究问题。 + +最后停在一页二维码/路径：PR、`issue128-certificate.json`、 +30 页长论文 PDF、技术报告 PDF、验证命令。 + +## 问答备忘 + +- **为什么不是 5×？** 因为 `r≤78` 的完整全局误差账本尚未闭合；D5 或 D8 的单项改善不能代替总证书。 +- **启发式分组会不会不严谨？** 搜索可启发式，接受条件不是；verifier 精确检查每个成员、覆盖关系和每一对反对易。 +- **为什么可把无限格点密度用于 12×12？** 证书的平移规范化、支撑半径和有限尺寸条件由 verifier 检查；不靠未经检查的热力学极限外推。 +- **倍率是否改变了门数模型？** 否。对照与本工作使用同一 + `G(r)=30r+1` 合并规律和同一 bond/CNOT 上界。 +- **浮点数在哪里出现？** 只用于排序、显示和小型稠密 sanity check；接受/拒绝使用整数、有理数和向外区间。 diff --git a/tracks/qcs/solutions/WangTheoPhys/issue128/docs/report/ERRATA.md b/tracks/qcs/solutions/WangTheoPhys/issue128/docs/report/ERRATA.md new file mode 100644 index 000000000..824a9fd07 --- /dev/null +++ b/tracks/qcs/solutions/WangTheoPhys/issue128/docs/report/ERRATA.md @@ -0,0 +1,49 @@ +# Erratum for the frozen short technical report + +Date recorded: 2026-07-31 (Asia/Shanghai) + +## Scope + +The six-page short report and the reviewer README frozen at commit +`ebb055926302c7d911e5ab00805513c382ac458d` contain a legacy auxiliary table +for recursive Suzuki orders 2, 4, 6, and 8. That table does not agree with the +authoritative schema-v3 main certificate. + +The affected legacy values are: + +```text +269,677; 36,361; 83,101; 377,251 merged groups +``` + +The authoritative values stored in +`certificates/issue128-certificate.json` under +`published_recursive_suzuki_audit` are: + +| Recursive order | Steps | Stages per step | Merged groups | Error upper summary | +|---:|---:|---:|---:|---:| +| 2 | 29,964 | 7 | 179,785 | `9.9999203748666826e-7` | +| 4 | 808 | 31 | 24,241 | `9.9682621337975879e-7` | +| 6 | 370 | 151 | 55,501 | `9.8463753882538689e-7` | +| 8 | 335 | 751 | 251,251 | `9.9829085642259163e-7` | + +## Authority rule + +For exact numerical claims, the JSON certificate and its SHA-256-bound +sidecars are authoritative. The old short-report table is retained unchanged +so that the frozen manifest remains reproducible; it must not be cited as +current evidence. The long-form manuscript uses the schema-v3 values. + +## Impact on the principal result + +There is no impact on the certified result. The auxiliary recursive-order +audit is not used to derive any of the following primary quantities: + +- pinned control: 393 steps and 11,791 merged groups; +- accepted candidate: 97 steps and 2,911 merged groups; +- adjacent boundary: 97 accepted and 96 rejected; +- exact resource ratio: `11791/2911 = 4.050498110614909...`; +- D4 partition: 75,324 terms in 7,576 verified groups. + +The primary quantities remain bound to the unchanged main certificate with +SHA-256 +`0a09623ce3b292a3637065c870fb3153bbdcddce30aef968565c4db3ddfc7201`. diff --git a/tracks/qcs/solutions/WangTheoPhys/issue128/docs/report/issue128-hpc-sprint-log.md b/tracks/qcs/solutions/WangTheoPhys/issue128/docs/report/issue128-hpc-sprint-log.md new file mode 100644 index 000000000..42ca9148b --- /dev/null +++ b/tracks/qcs/solutions/WangTheoPhys/issue128/docs/report/issue128-hpc-sprint-log.md @@ -0,0 +1,53 @@ +# Issue 128 六小时 HPC 冲刺运行记录 + +记录日期：2026-07-31（Asia/Shanghai） + +## 冻结主张 + +本轮计算不覆盖既有认证结果。当前唯一可对外报告的倍率仍为 +`11791/2911 = 4.050498110614909...`，对应 `r=393 → r=97`。除非新的完整全局误差账本及 fast/deep verifier 同时通过，否则不得把 D8 分项结果宣传为新的整体倍率。 + +## 可复现源状态 + +- 本地分支：`codex/issue128-hpc-six-hour` +- 生产源提交：`e65c95f4ae76dc1a78e81715cac21cfd7a81dbf8` +- 远端入口：`xh5-acamtw` +- 远端仓库：`/work/home/acamtw70yu/quantum-harness-issue128-hpc` +- Slurm account/QOS/partition：`giggleliu` / `user_acamtw70yu` / `xhacnormalb` +- 计算类型：CPU 上的 Python 精确整数/有理数与稀疏 Pauli 字典；GPU 不适用。 + +## 快速门槛 + +- Python 文件通过 `py_compile`。 +- 坐标无关制品、确定性 gzip、manifest 与损坏拒绝测试：3 passed。 +- 本地 stage 30 / order 8 smoke 成功，重复制品哈希稳定。 +- 本地常规测试：93 passed，11 deselected，41.12 秒。 +- 本地 deep verifier：`valid=true`，`deep_proof_regenerated=true`。 +- 远端 smoke：作业 `23044170`，stages 28--30 完成且 manifest/hash 正常；stage 27 用作较重梯度检查。 + +## 生产图 + +| 角色 | Slurm job | 资源/上限 | 依赖 | +|---|---:|---|---| +| 31-stage exact D8 array | `23044178` | 每单元 17 CPU、64 GiB、4:30:00 | 无 | +| deep verifier + 全测试 | `23044196` | 9 CPU、32 GiB、4:30:00 | 无 | +| exact reducer | `23044212` | 26 CPU、96 GiB、1:15:00 | `afterok:23044178` | + +数组最初以 16 路并发启动；确认账号/QOS允许后，用 Slurm 正常调度接口把 `ArrayTaskThrottle` 提升到 31。所有剩余单元随后进入 RUNNING；调度和授权仍由集群控制器执行。 + +## 制品契约 + +每个 `stage-XX` 目录包含： + +- `shard.json.gz`：坐标寻址、registry 无关、canonical JSON、确定性 gzip； +- `manifest.json`：stage/order、开始结束时间、wall time、峰值 RSS、term counts、源提交、dirty 标志和 SHA-256； +- Slurm stdout/stderr：启动与完成状态。 + +归并器要求 31 个 manifest 全部成功、公式 ID/order/source commit 一致、每个 shard 哈希匹配，并要求正向与反向精确归并得到相同字典。任何失败均不得生成倍率主张。 + +## 终局判定 + +1. 若数组和 reducer 完成：读取 exact D8 site-density upper，重建 `r=78` 的完整 D4--D8+账本，再运行 fast/deep verifier。 +2. 若任一 stage OOM/timeout：保留成功制品和 sacct 证据，按失败类别只重提缺失单元，不把部分和当全局界。 +3. 若完整账本 `≤10⁻⁶`：可以进入五倍候选复核；仍需独立证书、报告和 fresh-clone 重现。 +4. 若完整账本 `>10⁻⁶`：报告严格负结果与缺口，主倍率保持 4.050498×。 diff --git a/tracks/qcs/solutions/WangTheoPhys/issue128/docs/report/issue128-hpc-sprint-report.tex b/tracks/qcs/solutions/WangTheoPhys/issue128/docs/report/issue128-hpc-sprint-report.tex new file mode 100644 index 000000000..f37a94742 --- /dev/null +++ b/tracks/qcs/solutions/WangTheoPhys/issue128/docs/report/issue128-hpc-sprint-report.tex @@ -0,0 +1,145 @@ +\documentclass[10pt]{article} +\usepackage[margin=0.72in]{geometry} +\usepackage[T1]{fontenc} +\usepackage{lmodern} +\usepackage{microtype} +\usepackage{amsmath,amssymb} +\usepackage{booktabs} +\usepackage{xcolor} +\usepackage{hyperref} +\hypersetup{ + colorlinks=true, + urlcolor=blue!60!black, + linkcolor=blue!55!black, + pdftitle={Issue 128 Exact-D8 HPC Sprint: Interim Technical Supplement}, + pdfauthor={Junkai Wang (WangTheoPhys)} +} +\setlength{\parindent}{0pt} +\setlength{\parskip}{0.30em} + +\title{Issue 128 Exact-D8 HPC Sprint\\\large Interim Technical Supplement} +\author{Junkai Wang (WangTheoPhys)} +\date{31 July 2026} + +\begin{document} +\maketitle + +\begin{abstract} +This supplement records a six-hour, proof-directed HPC sprint following the +machine-certified Issue 128 result. The accepted claim remains frozen at +$11791/2911=4.050498110614909\ldots$; no fivefold result is asserted here. +The sprint computes the order-eight coefficient of the exact right generator +as 31 independently checkable stage shards, runs the accepted deep verifier +independently, and ranks 24 matching-color orders in an explicitly untrusted +discovery path. Every production artifact is atomic, hash-addressed, and +bound to a source commit. This is an interim execution report: the final +claim gate remains closed until all shards are reduced and a complete global +error ledger passes both fast and deep verification. +\end{abstract} + +\section{Frozen result and research gate} + +For the periodic $12\times12$ isotropic Heisenberg benchmark at $T=1$ and +global operator-norm tolerance $10^{-6}$, the certified fourth-order Suzuki +result accepts $r=97$ and rejects $r=96$. With $G(r)=30r+1$ merged group +exponentials, the exact resource ratio is +\[ + \frac{G(393)}{G(97)}=\frac{11791}{2911} + =4.050498110614909\ldots . +\] +Fivefold arithmetic would require $r\le78$, but this arithmetic is not an +error proof. The present experiment targets one missing ingredient: a +concrete exact order-eight right-generator coefficient map. A new multiplier +may be reported only if a complete D4--D8-plus global ledger at the proposed +step count is at most $10^{-6}$ and independent verification succeeds. + +\section{Exact sharding algorithm} + +For stages $V_j(t)=\exp(-i a_j t H_{c_j})$ in the fixed 31-stage formula, +linearity gives a decomposition of the right generator into 31 contributions. +Each array task starts from $a_jH_{c_j}$ and conjugates it through all later +stages, truncating only the formal series degree at eight. Coefficients live +exactly in $\mathbb{Q}[\alpha]/(\alpha^3-4)$; Pauli operators are canonicalized +under the $2\times2$ translation cell before serialization. + +The shard format does not expose process-local registry bit indices. Every +Pauli string is encoded by physical integer coordinates and labels. Canonical +JSON and gzip with fixed metadata make repeated output byte-deterministic. +The manifest records stage, order, timestamps, term counts, peak RSS, source +commit, dirty flag, and SHA-256. The reducer rejects a missing stage, wrong +formula/order, mixed source state, corrupt hash, or duplicate stage. It also +requires forward and reverse exact reduction to produce identical maps. + +For a cubic coefficient $c=a_0+a_1\alpha+a_2\alpha^2$, an outward rational +interval for $\alpha=\sqrt[3]{4}$ encloses $c$. Summing the larger endpoint +magnitudes yields a rigorous Pauli-$\ell_1$ upper bound. This bound is a +screening ingredient; it is not by itself the global $r=78$ certificate. + +\section{Execution contract} + +The production source is commit +\texttt{e65c95f4ae76dc1a78e81715cac21cfd7a81dbf8}. Jobs run on the +\texttt{xhacnormalb} CPU partition under account \texttt{giggleliu} and QOS +\texttt{user\_acamtw70yu}. GPU partitions are intentionally excluded because +the workload is sparse Python integer/rational algebra. + +\begin{table}[h] +\centering +\begin{tabular}{@{}lllr@{}} +\toprule +Role & Slurm job & Per-task allocation & Limit \\ +\midrule +Exact D8 array (31 stages) & 23044178 & 17 CPU, 64 GiB & 4:30 \\ +Independent verification & 23044196 & 9 CPU, 32 GiB & 4:30 \\ +Exact reducer & 23044212 & 26 CPU, 96 GiB & 1:15 \\ +Order screen (24 permutations) & 23044214 & 5 CPU, 16 GiB & 1:00 \\ +\bottomrule +\end{tabular} +\caption{Submitted Slurm work. The reducer has an \texttt{afterok} +dependency on the D8 array.} +\end{table} + +The array first passed \texttt{sbatch --test-only}. A four-cell remote smoke +then completed the lightest stages and produced matching deterministic +artifacts. The partition's 3931 MiB-per-CPU memory rule was discovered by the +test-only gate and reflected in all allocations. Once production started, +the voluntary array throttle was raised from 16 to 31; Slurm and QOS retained +authority over placement and admission. + +\section{Verification evidence at launch} + +The local proof-preservation gate reported 93 passing tests and 11 deselected +slow tests in 41.12 seconds. A separate deep verifier run returned +\texttt{valid=true} and \texttt{deep\_proof\_regenerated=true}, reproduced +$393\to97$, and returned the exact ratio \texttt{11791/2911}. At the first +full-concurrency snapshot, the D8 run contained 28 running manifests, three +complete manifests, and zero failures. The light-stage completion times were +approximately 5.61 seconds (stage 28), 0.171 seconds (stage 29), and 0.024 +seconds (stage 30). These timings validate the transport and serialization +path; they do not predict the much heavier early stages. + +The matching-order array is a separate discovery lane. Its JSON records +\texttt{trusted=false} and states that its purpose is discovery ranking. A +favorable ranking can nominate a formula for exact reconstruction but cannot +enter the certificate directly. + +\section{Final adjudication protocol} + +The outcome will be classified as follows. +\begin{enumerate} + \item If all 31 shards and exact reduction succeed, insert the resulting D8 + upper bound into a complete $r=78$ ledger and rerun fast/deep checks. + \item If any task reaches OOM or walltime, retain successful shards and + scheduler evidence, retry only missing cells if time remains, and make + no inference from a partial sum. + \item If the complete ledger is at most $10^{-6}$, prepare a new independently + bound certificate before describing a fivefold result. + \item Otherwise publish the strict negative result or remaining numerical + gap, while keeping the certified 4.050498-fold result unchanged. +\end{enumerate} + +\textbf{Interim status.} This document records a running experiment, not a +new performance claim. The accepted result remains 4.050498-fold until the +final gate above is closed. + +\end{document} diff --git a/tracks/qcs/solutions/WangTheoPhys/issue128/docs/report/issue128-technical-report.tex b/tracks/qcs/solutions/WangTheoPhys/issue128/docs/report/issue128-technical-report.tex new file mode 100644 index 000000000..2fffc0391 --- /dev/null +++ b/tracks/qcs/solutions/WangTheoPhys/issue128/docs/report/issue128-technical-report.tex @@ -0,0 +1,359 @@ +\documentclass[11pt]{article} + +\usepackage[margin=0.82in]{geometry} +\usepackage[T1]{fontenc} +\usepackage[utf8]{inputenc} +\usepackage{lmodern} +\usepackage{microtype} +\usepackage{amsmath,amssymb} +\usepackage{booktabs} +\usepackage{array} +\usepackage{tabularx} +\usepackage{xcolor} +\usepackage{hyperref} +\usepackage{url} + +\hypersetup{ + colorlinks=true, + linkcolor=blue!55!black, + citecolor=blue!55!black, + urlcolor=blue!60!black, + pdftitle={Issue 128: A Machine-Certified 4.050498x Trotter Resource Reduction}, + pdfauthor={Junkai Wang (WangTheoPhys)} +} + +\setlength{\parindent}{0pt} +\setlength{\parskip}{0.45em} +\setlength{\emergencystretch}{2em} +\newcommand{\opnorm}[1]{\left\lVert #1\right\rVert} +\newcommand{\PauliLOne}{\ell_{1,\mathrm{Pauli}}} + +\title{\textbf{Issue 128: A Machine-Certified 4.050498$\times$\\ +Trotter Resource Reduction for a 2D Heisenberg Benchmark}} +\author{Junkai Wang (WangTheoPhys)\\ +\small Quantum Circuit Simulation Track} +\date{July 30, 2026} + +\begin{document} +\maketitle + +\begin{abstract} +We give an exact-rational, independently checkable certificate for simulating +the periodic $12\times12$ spin-$1/2$ isotropic Heisenberg model to global +operator-norm error at most $10^{-6}$ for time $T=1$. The circuit is the same +five-copy fourth-order Suzuki product formula, with the same four matching +fragments and the same compilation cost model, on both sides of the +comparison. A pinned instantiation of the published high-order commutator +bound requires 393 steps and 11,791 merged group exponentials. Our certificate +accepts 97 steps and rejects 96, requiring 2,911 merged group exponentials. +The exact improvement is $11791/2911=4.050498110614909\ldots$, with a +75.3117\% resource reduction. The gain comes from delaying norm inequalities: +we compute the formula logarithm, project it to Lie elements, evaluate the +concrete lattice representation, combine identical Pauli strings, and certify +pairwise-anticommuting groups before closing a finite-step right-generator +ledger with a rational tail. The result exceeds the challenge's twofold +requirement by a factor of 2.025. An exact degree-five side study is also +included, but no 78-step global error certificate is claimed or supplied. +\end{abstract} + +\section{Claim boundary and benchmark} + +The certified benchmark is +\begin{equation} + H=\sum_{\langle i,j\rangle} + \frac{X_iX_j+Y_iY_j+Z_iZ_j}{4}, + \qquad L=12,\quad N=L^2=144,\quad T=1, + \label{eq:hamiltonian} +\end{equation} +with periodic boundary conditions and target +$\opnorm{U(T)-\widetilde U(T)}\le 10^{-6}$. Horizontal and vertical bonds are +split by parity into four commuting matchings. A symmetric second-order step +is composed five times to obtain the standard fourth-order Suzuki formula. +After adjacent equal fragments are merged, an $r$-step simulation contains +\begin{equation} + G(r)=30r+1 + \label{eq:groups} +\end{equation} +group exponentials. Each group has $N/2=72$ disjoint bond propagators, and we +use the same upper bound of three CNOT gates per Heisenberg bond propagator on +both sides. + +This report makes one principal claim: the supplied exact certificate proves +that $r=97$ is sufficient, while its own error function rejects $r=96$. The +comparison fixes the Hamiltonian, product formula, fragmentation, time, +tolerance, and cost model. It therefore measures a proof/certification gain, +not a change in physical problem or a favorable redefinition of cost. + +The published control is an interval instantiation of the high-order +commutator framework of Childs, Su, Tran, Wiebe, and Zhu +\cite{childs2021theory}, using the expansion-first local Pauli-$\ell_1$ +procedure also connected to later commutator-scaling work +\cite{schubert2023trotter}. All 31 admissible theorem centers were scanned; +center 20 gives the lowest pinned control, 393 steps. This control is stored +in the main certificate rather than recomputed from a floating-point constant. + +\section{Where the improvement comes from} + +Let the four anti-Hermitian matching generators be $A_1,\ldots,A_4$, and let +$S_4(t)$ denote one merged fourth-order step. A conventional bound expands +partial Hamiltonian sums into many nested commutators and applies triangle +inequalities early. That destroys cancellations before the concrete +Heisenberg algebra is visible. Our method reverses this order: + +\begin{enumerate} + \item compute the homogeneous free-associative-word logarithm of the complete + 31-stage formula; + \item use the Dynkin--Specht--Wever map to project the degree-five and + degree-seven pieces into their Lie elements; + \item evaluate those Lie elements in the four-matching Heisenberg + representation on the infinite translation cell; + \item canonicalize translations and combine identical Pauli strings before + any final norm inequality; + \item partition the leading right-generator terms into exactly verified + pairwise-anticommuting groups; and + \item convert the logarithm to the right generator, retain degrees four + through seven explicitly, and bound the remaining degrees by a + rational geometric locality tail. +\end{enumerate} + +If the Pauli strings in $K_g=\sum_j c_jP_j$ anticommute pairwise, then +$K_g^2=(\sum_jc_j^2)I$, and hence +\begin{equation} + \opnorm{K_g}=\sqrt{\sum_j |c_j|^2}. + \label{eq:anticommute} +\end{equation} +We use an outward rational upper bound for each square root. The greedy +grouping algorithm is only a discovery tool: the sidecar records every term, +every group membership, and every group bound. The verifier independently +checks exact coverage and every symplectic anticommutation relation. Thus a +different grouping heuristic can change efficiency, but it cannot weaken the +soundness of an accepted certificate. + +The submitted leading-defect sidecar contains 75,324 canonical Pauli terms in +7,576 groups, with maximum group size 10. The direct termwise cell bound +20.160968407335066 is reduced to the certified grouped bound +6.472926505087888, an improvement of 3.114660484942633 at the leading local +norming step. + +\section{A Heisenberg-specific local lemma} + +The higher-degree ledger also benefits from an exact local observation. For +one normalized bond +\begin{equation} + h=\frac{XX+YY+ZZ}{4}, +\end{equation} +every phase-free two-qubit Pauli string anticommutes with either zero or +exactly two of $XX$, $YY$, and $ZZ$. Direct enumeration of the 16 local +Pauli cases therefore gives +\begin{equation} + \opnorm{[h,P]}_{\PauliLOne} + \le \opnorm{P}_{\PauliLOne}, + \label{eq:locallemma} +\end{equation} +so the bond commutator growth constant is 1 rather than the generic +triangle-inequality value $3/2$. This is a small lemma with a large cumulative +effect because it enters repeated support-growth estimates. The verifier +checks the complete local table exactly; it does not assume the lemma as an +external analytic input. + +This separation of discovery and trust is important. Symbolic search finds a +compact representation and a strong grouping, while the trusted path uses +integers, rational numbers, outward rational intervals, and exact symplectic +Pauli multiplication. Floating point is used only to rank candidates, print +human-readable decimals, and perform a small dense sanity check. + +\section{Finite-step error certificate} + +Write the one-step logarithm as +\begin{equation} + \log S_4(t)=tA+t^5B+t^7C+O(t^9), +\end{equation} +where $A=\sum_{j=1}^4 A_j$. The exact right logarithmic generator has the +low-degree expansion +\begin{align} + S_4'(t)S_4(t)^{-1}-A + ={}&5t^4B+2t^5[A,B] \\ + &+t^6\!\left(7C+\frac23[A,[A,B]]\right) \\ + &+t^7\!\left(3[A,C]+\frac16[A,[A,[A,B]]]\right) + +R_{\ge8}(t). + \label{eq:rightgenerator} +\end{align} +The D4 term is evaluated with the exact anticommuting partition. Degrees D5 +through D7 are bounded by the certified concrete coefficients and the local +lemma. Support growth, conjugation Taylor factorials, and the Duhamel +integration factor make $R_{\ge8}$ an explicit rational geometric series. +Nothing beyond degree seven is silently discarded. + +For $r=97$, the exact rational contributions have the following outward +decimal renderings. + +\begin{table}[ht] +\centering +\caption{Certified global operator-norm error ledger at $r=97$.} +\label{tab:errorledger} +\begin{tabular}{@{}lr@{}} +\toprule +Right-generator contribution & Global upper bound \\ +\midrule +Degree 4 & $5.264367936822258\times10^{-7}$ \\ +Degree 5 & $1.383487536744612\times10^{-7}$ \\ +Degree 6 & $1.579057000594835\times10^{-7}$ \\ +Degree 7 & $1.894657080691470\times10^{-8}$ \\ +Degree 8 and above & $1.542560312083473\times10^{-7}$ \\ +\midrule +Total & $9.958938494314325\times10^{-7}$ \\ +\bottomrule +\end{tabular} +\end{table} + +The underlying total is a rational number and is strictly below $10^{-6}$. +At $r=96$, the outward decimal is +$1.050565873970784\times10^{-6}$, strictly above the target. The step boundary +is checked by cross-multiplying integers, not by comparing rounded decimals. +The printed decimal convention rounds positive certified upper bounds upward +at 16 significant digits, so display conversion never understates the bound. + +\section{Resource result} + +Because both rows use the same 31-stage formula and compilation model, the +resource comparison follows directly from Eq.~\eqref{eq:groups}. + +\begin{table}[ht] +\centering +\caption{Pinned rigorous control and certified result.} +\label{tab:resources} +\begin{tabular}{@{}lrrr@{}} +\toprule +Resource & Published control & This work & Reduction \\ +\midrule +Trotter steps & 393 & 97 & 75.32\% \\ +Merged group exponentials & 11,791 & 2,911 & 75.31\% \\ +Bond propagators & 848,952 & 209,592 & 75.31\% \\ +CNOT upper bound & 2,546,856 & 628,776 & 75.31\% \\ +\bottomrule +\end{tabular} +\end{table} + +The primary exact ratio is +\begin{equation} + \frac{G(393)}{G(97)}=\frac{11791}{2911} + =4.050498110614909\ldots. +\end{equation} +The challenge asks for at least a twofold resource improvement. The achieved +ratio is 2.025249 times that requirement---equivalently, 202.5\% of the stated +minimum ratio, or 102.5\% beyond it. We call this a fourfold certified result, +not a fivefold result. + +\section{Independent checks and negative results} + +The certificate is designed to be reviewable without trusting the search +code. The fast verifier checks the main certificate digest bindings, all +75,324 D4 terms, exact coverage by 7,576 groups, every pairwise +anticommutation relation, each rational square-root upper bound, the integer +step boundary, and all resource arithmetic. A deep mode regenerates the +published control, the canonical leading-defect coefficients, the +degree-seven majorant, and the finite tail from the formula. + +A degenerate $2\times2$ periodic-algebra cross-check gives actual dense +spectral-norm error $9.128277711816321\times10^{-10}$, below its outward +certificate $2.766666666666667\times10^{-8}$. This is not used to infer the +$12\times12$ result; it catches representation, stage-ordering, and +normalization errors on a tractable instance. + +We also record routes that did not justify a stronger claim: +\begin{itemize} + \item The degree-three Lie representation has exact rank 20, equal to the + free-Lie dimension $(4^3-4)/3=20$. The four matching fragments thus + offer no free model-specific reduction of fourth-order conditions. + \item Standard recursive Suzuki orders 2, 4, 6, and 8 compile to 269,677, + 36,361, 83,101, and 377,251 groups, respectively, under the audited + generic locality-tail bound. Raising recursive order alone is worse. + \item A three-site SU(2) cluster decomposition improves local block counts + but not the conservative compiled CNOT bound after including the + three-site propagator cost. + \item Empirically optimized formulas were not substituted for a deterministic + worst-case certificate. Their sampled constants cannot replace an + interval-certified comparison on this benchmark. +\end{itemize} + +\section{Fivefold feasibility boundary} + +Fivefold arithmetic would require $G(r)\le\lfloor11791/5\rfloor$, hence +$r\le78$. Conditional on an accepted 78-step error proof, +$G(78)=2341$ and $11791/2341=5.036736437419906\ldots$. That arithmetic is not +an error certificate. + +As a concrete follow-up, we generated and independently verified an exact +degree-five sidecar with 605,832 canonical Pauli terms, 123,106 same-support +pairwise-anticommuting groups, and site-density upper bound +11.23706750025696. This closes one useful algebraic subproblem and validates +the streamed cubic-field machinery. It does not close the global ledger at +$r=78$: the translation-coupled D4 budget and explicit D8-plus delayed-tail +certificate remain hard gates. \textbf{No 78-step global error certificate is +claimed or supplied.} + +The audit therefore gives a precise next research decision. Rebuild the +complete exact ledger at $r=78$ only after D4 and D8-plus gates are closed; if +the sum is at most $10^{-6}$, fivefold can be claimed. For tenfold, the +current fourth-order ledger is off by hundreds even with the D4 contribution +removed, so an effective order lift---for example a certified processor or a +shorter sixth-order kernel---is required. + +\section{Reproduction and delivered evidence} + +From the \texttt{issue128} directory, a reviewer can run +\begin{verbatim} +python -m pip install -e '.[test]' +pytest -q +PYTHONPATH=src python scripts/verify.py \ + certificates/issue128-certificate.json +PYTHONPATH=src python scripts/build_d5_certificate.py --verify-only +PYTHONPATH=src python scripts/package_delivery.py --check +shasum -a 256 -c artifacts/SHA256SUMS +\end{verbatim} + +The delivery contains: (i) the main exact-rational certificate; (ii) D4 and +D5 sidecars bound by SHA-256; (iii) machine-readable JSON and plain-text result +summaries; (iv) a captured verification transcript with exit codes; (v) this +report and its source; (vi) regression tests; and (vii) one checksum manifest +binding all reviewer-facing evidence. The repository and pull request are the +transport mechanism; the numerical claim is anchored in the exact artifacts +and executable verification path. + +The three large proof inputs have fixed identities: +\begin{center} +\small +\begin{tabularx}{\textwidth}{@{}lX@{}} +\toprule +Artifact & SHA-256 \\ +\midrule +Main certificate & +\texttt{0a09623ce3b292a3637065c870fb3153bbdcddce30aef968565c4db3ddfc7201} \\ +D4 sidecar & +\texttt{a397414bb0229fb1ebdb38798aa781fb89dbb9d5cdbed94c7cd2e9120da62718} \\ +D5 sidecar & +\texttt{c5e8968a93b4497b41fe42c0e364324388272e183e1fcd20a536bc988f5361dd} \\ +\bottomrule +\end{tabularx} +\end{center} +The committed \texttt{artifacts/SHA256SUMS} file additionally binds the final +PDF, report source, summaries, transcript, cross-check, and sidecars as one +reviewer-facing package. + +\section{Conclusion} + +Issue 128's basic requirement is fully met for the fixed benchmark. The +certified resource count drops from 11,791 to 2,911 merged group exponentials, +an exact 4.050498-fold improvement, without changing the circuit family or +cost model. The central algorithmic contribution is not merely a faster +implementation: it is a tighter proof architecture that preserves concrete +Lie/Pauli cancellation until exact local algebra and anticommutation can be +used, then closes the remaining finite-step error with rational arithmetic. +The supplied data, transcript, tests, and PDF are mutually hash-bound and +independently reproducible. Fivefold remains a clearly delimited follow-up, +not part of the certified result. + +\bibliographystyle{unsrt} +\bibliography{references} + +\end{document} diff --git a/tracks/qcs/solutions/WangTheoPhys/issue128/docs/report/output/issue128-technical-report.pdf b/tracks/qcs/solutions/WangTheoPhys/issue128/docs/report/output/issue128-technical-report.pdf new file mode 100644 index 000000000..446e71f2e Binary files /dev/null and b/tracks/qcs/solutions/WangTheoPhys/issue128/docs/report/output/issue128-technical-report.pdf differ diff --git a/tracks/qcs/solutions/WangTheoPhys/issue128/docs/report/output/pdf/issue128-hpc-sprint-report.pdf b/tracks/qcs/solutions/WangTheoPhys/issue128/docs/report/output/pdf/issue128-hpc-sprint-report.pdf new file mode 100644 index 000000000..f43c8b05e Binary files /dev/null and b/tracks/qcs/solutions/WangTheoPhys/issue128/docs/report/output/pdf/issue128-hpc-sprint-report.pdf differ diff --git a/tracks/qcs/solutions/WangTheoPhys/issue128/docs/report/references.bib b/tracks/qcs/solutions/WangTheoPhys/issue128/docs/report/references.bib new file mode 100644 index 000000000..51de16e91 --- /dev/null +++ b/tracks/qcs/solutions/WangTheoPhys/issue128/docs/report/references.bib @@ -0,0 +1,24 @@ +@article{childs2021theory, + title = {A Theory of Trotter Error}, + author = {Childs, Andrew M. and Su, Yuan and Tran, Minh C. and Wiebe, Nathan and Zhu, Shuchen}, + journal = {Physical Review X}, + volume = {11}, + number = {1}, + pages = {011020}, + year = {2021}, + doi = {10.1103/PhysRevX.11.011020}, + eprint = {1912.08854}, + archivePrefix = {arXiv}, + primaryClass = {quant-ph} +} + +@article{schubert2023trotter, + title = {Trotter Error with Commutator Scaling for the {Fermi--Hubbard} Model}, + author = {Schubert, Ansgar and Mendl, Christian B.}, + journal = {arXiv preprint arXiv:2306.10603}, + year = {2023}, + eprint = {2306.10603}, + archivePrefix = {arXiv}, + primaryClass = {quant-ph}, + url = {https://arxiv.org/abs/2306.10603} +} diff --git a/tracks/qcs/solutions/WangTheoPhys/issue128/docs/superpowers/plans/2026-07-29-anticommuting-d4-certificate.md b/tracks/qcs/solutions/WangTheoPhys/issue128/docs/superpowers/plans/2026-07-29-anticommuting-d4-certificate.md new file mode 100644 index 000000000..2be42d34d --- /dev/null +++ b/tracks/qcs/solutions/WangTheoPhys/issue128/docs/superpowers/plans/2026-07-29-anticommuting-d4-certificate.md @@ -0,0 +1,948 @@ +# Anticommuting D4 Certificate Implementation Plan + +> **For agentic workers:** REQUIRED SUB-SKILL: Use superpowers:subagent-driven-development (recommended) or superpowers:executing-plans to implement this plan task-by-task. Steps use checkbox (`- [ ]`) syntax for tracking. + +**Goal:** Replace the Pauli-l1 bound on the local degree-four right-generator defect with an exact-rational pairwise-anticommuting partition certificate, certify 98 Trotter steps, and update PR #248 to a resource improvement above 4x. + +**Architecture:** The existing interval Lie-series code will expose a canonical `2x2`-unit-cell D4 coefficient map. A deterministic optimizer will partition those Pauli terms into pairwise-anticommuting groups, while a separate proof function and verifier check coverage, symplectic anticommutation, and outward rational Euclidean bounds. A compact sidecar stores the large proof object; the main schema-v3 certificate binds it by SHA-256 and uses only the grouped D4 bound while retaining the existing D5-D7 and tail bounds. + +**Tech Stack:** Python 3.12, `Fraction`, existing `RationalInterval`, exact binary symplectic Pauli arithmetic, JSON, SHA-256, pytest. + +## Global Constraints + +- Benchmark remains the periodic `12x12` isotropic spin-1/2 Heisenberg Hamiltonian with `h_ij=(XX+YY+ZZ)/4`, `T=1`, and operator-norm tolerance `1e-6`. +- Formula remains the 31-stage five-copy fourth-order Suzuki formula over the same four disjoint bond matchings. +- Cost remains `30*r+1` merged group exponentials, `N/2` bond propagators per group, and three CNOTs per bond propagator. +- The proof path uses exact integers, rational arithmetic, rational intervals, and outward rounding; floating point may choose a partition but may not establish an inequality. +- The previous 116-step certificate remains reproducible until the new 98-step certificate passes fast verification, deep regeneration, solution tests, repository tests, and the dense small-size cross-check. +- All committed files stay under `tracks/qcs/solutions/WangTheoPhys/`. + +--- + +## File Structure + +- Modify `src/trottercert/refined_error.py`: construct canonical interval D4 coefficients and accept a grouped D4 site bound. +- Create `src/trottercert/anticommuting.py`: deterministic partition discovery and exact partition certification. +- Modify `src/trottercert/verify.py`: verify the D4 sidecar in fast mode and regenerate it in deep mode. +- Modify `scripts/build_v3_certificate.py`: build the grouped proof, write the compact sidecar, bind its hash, and search the new integer boundary. +- Modify `src/trottercert/crosscheck.py`: update the outward summary and dense check to the certified step count. +- Create `certificates/issue128-d4-groups.json`: compact exact proof object. +- Modify `certificates/issue128-certificate.json`: schema-v3 candidate fields and 98-step resources. +- Modify `certificates/issue128-small-crosscheck.json`: dense check at 98 steps. +- Modify `certificates/issue128-verification.txt`: deep-verification transcript and resource ledger. +- Modify `README.md`: explain the D4 theorem and report the new count. +- Modify `tests/test_refined_error.py`, `tests/test_resources_verify.py`, and `tests/test_small_crosscheck.py`. +- Create `tests/test_anticommuting.py`. + +--- + +### Task 1: Canonical exact D4 coefficient map + +**Files:** +- Modify: `src/trottercert/refined_error.py` +- Test: `tests/test_refined_error.py` + +**Interfaces:** +- Produces: `certified_d4_cell_coefficients(stages: Sequence[IntervalStage], *, quantization_digits: int = 18) -> dict[SymplecticPauli, RationalInterval]`. +- Preserves: `certified_leading_e5_cell_l1(stages: Sequence[IntervalStage], *, quantization_digits: int = 24) -> Fraction` and its current ungrouped bound for repeated-adjoint estimates. + +- [ ] **Step 1: Write a failing canonicalization test** + +Add a test that constructs two translated symplectic Pauli terms in one shared `CoordinateRegistry`, canonicalizes them with the colored unit cell, and requires identical output: + +```python +def test_colored_unit_cell_canonicalization_merges_translates() -> None: + registry = CoordinateRegistry() + left = symplectic_pauli_from_coordinates( + registry, ((0, 0, "X"), (1, 0, "X")) + ) + right = symplectic_pauli_from_coordinates( + registry, ((2, 0, "X"), (3, 0, "X")) + ) + assert canonicalize_symplectic_unit_cell(registry, left) == ( + canonicalize_symplectic_unit_cell(registry, right) + ) +``` + +- [ ] **Step 2: Run the focused test and verify failure** + +Run: + +```bash +pytest -q tests/test_refined_error.py::test_colored_unit_cell_canonicalization_merges_translates +``` + +Expected: import failure because `canonicalize_symplectic_unit_cell` and `symplectic_pauli_from_coordinates` do not exist. + +- [ ] **Step 3: Implement exact symplectic translation canonicalization** + +Add these signatures to `refined_error.py`: + +```python +def symplectic_pauli_from_coordinates( + registry: CoordinateRegistry, + coordinates: Sequence[tuple[int, int, str]], +) -> SymplecticPauli: + x_mask = z_mask = 0 + for x, y, op in coordinates: + bit = 1 << registry.site((x, y)) + if op in {"X", "Y"}: + x_mask |= bit + if op in {"Z", "Y"}: + z_mask |= bit + return x_mask, z_mask + + +def canonicalize_symplectic_unit_cell( + registry: CoordinateRegistry, + pauli: SymplecticPauli, + unit_cell: tuple[int, int] = (2, 2), +) -> SymplecticPauli: + x_mask, z_mask = pauli + sites = x_mask | z_mask + coordinates = [] + while sites: + bit = sites & -sites + site = bit.bit_length() - 1 + x = bool(x_mask & bit) + z = bool(z_mask & bit) + op = "Y" if x and z else ("X" if x else "Z") + cx, cy = registry.coordinate(site) + coordinates.append((cx, cy, op)) + sites ^= bit + if not coordinates: + return 0, 0 + step_x, step_y = unit_cell + min_x = min(x for x, _, _ in coordinates) + min_y = min(y for _, y, _ in coordinates) + shift_x = -(min_x - min_x % step_x) + shift_y = -(min_y - min_y % step_y) + return symplectic_pauli_from_coordinates( + registry, + tuple( + (x + shift_x, y + shift_y, op) + for x, y, op in coordinates + ), + ) +``` + +Decode all occupied bit positions through `registry.coordinate(site)`, shift +the minimum coordinates to their residues modulo `(2,2)`, and rebuild the +masks through `registry.site(...)`. Empty Paulis remain `(0, 0)`. + +- [ ] **Step 4: Refactor the E5 interval accumulator into a reusable map** + +Add: + +```python +def _leading_e5_cell_interval_coefficients( + stages: Sequence[IntervalStage], + *, + quantization_digits: int, + canonicalize: bool, +) -> dict[SymplecticPauli, RationalInterval]: + logarithm = interval_formula_log_series(stages, 5) + grid = 10**quantization_digits + evaluator = SymplecticDyadicLocalDensityEvaluator( + shared_coordinates=True + ) + registry = evaluator.registries[0] + integer_coefficients: dict[SymplecticPauli, tuple[int, int]] = {} + for word, coefficient in logarithm[5].items(): + rounded = outward_quantize(coefficient, grid) + word_lower = rounded.lower.numerator * ( + grid // rounded.lower.denominator + ) + word_upper = rounded.upper.numerator * ( + grid // rounded.upper.denominator + ) + for raw_pauli, numerator in evaluator.evaluate(word).items(): + pauli = ( + canonicalize_symplectic_unit_cell(registry, raw_pauli) + if canonicalize + else raw_pauli + ) + lower, upper = integer_coefficients.get(pauli, (0, 0)) + if numerator >= 0: + lower += numerator * word_lower + upper += numerator * word_upper + else: + lower += numerator * word_upper + upper += numerator * word_lower + if lower or upper: + integer_coefficients[pauli] = lower, upper + else: + integer_coefficients.pop(pauli, None) + denominator = grid * 5 * (1 << 6) + return { + pauli: RationalInterval( + Fraction(lower, denominator), + Fraction(upper, denominator), + ) + for pauli, (lower, upper) in integer_coefficients.items() + } +``` + +Reuse the existing word-endpoint loop. Accumulate integer lower and upper +numerators, optionally canonicalizing each evaluated Pauli before merging, +then divide by the common denominator +`grid * 5 * (1 << 6)`. + +Implement: + +```python +def certified_d4_cell_coefficients( + stages: Sequence[IntervalStage], + *, + quantization_digits: int = 18, +) -> dict[SymplecticPauli, RationalInterval]: + return { + pauli: coefficient * 5 + for pauli, coefficient in _leading_e5_cell_interval_coefficients( + stages, + quantization_digits=quantization_digits, + canonicalize=True, + ).items() + } +``` + +Keep `certified_leading_e5_cell_l1` defined as the sum of absolute upper +endpoints from the noncanonical map so D5-D7 retain the prior proof. + +- [ ] **Step 5: Add exact regression assertions** + +Extend the slow refined-error test: + +```python +coefficients = certified_d4_cell_coefficients(stages) +assert coefficients +assert all(interval.lower <= interval.upper for interval in coefficients.values()) +assert sum( + interval.abs_upper() for interval in coefficients.values() +) <= 5 * certified_leading_e5_cell_l1(stages) +``` + +- [ ] **Step 6: Run the refined-error tests** + +Run: + +```bash +pytest -q tests/test_refined_error.py +``` + +Expected: all unmarked tests pass; slow regeneration remains deselected by the project configuration. + +- [ ] **Step 7: Commit Task 1** + +```bash +git add tracks/qcs/solutions/WangTheoPhys/issue128/src/trottercert/refined_error.py \ + tracks/qcs/solutions/WangTheoPhys/issue128/tests/test_refined_error.py +git commit -m "expose canonical interval D4 coefficients" +``` + +--- + +### Task 2: Deterministic anticommuting partition and exact proof + +**Files:** +- Create: `src/trottercert/anticommuting.py` +- Create: `tests/test_anticommuting.py` + +**Interfaces:** +- Consumes: the `dict[SymplecticPauli, RationalInterval]` from Task 1. +- Produces: `AnticommutingPartitionCertificate`, `discover_anticommuting_partition(...)`, and `certify_anticommuting_partition(...)`. + +- [ ] **Step 1: Write failing proof tests** + +Create `tests/test_anticommuting.py` with: + +```python +from fractions import Fraction + +import pytest + +from trottercert.anticommuting import ( + certify_anticommuting_partition, + discover_anticommuting_partition, + symplectic_anticommutes, +) +from trottercert.intervals import RationalInterval + + +def test_symplectic_anticommutation() -> None: + x = (1, 0) + y = (1, 1) + z = (0, 1) + assert symplectic_anticommutes(x, y) + assert symplectic_anticommutes(y, z) + assert symplectic_anticommutes(z, x) + + +def test_exact_group_bound_uses_euclidean_norm() -> None: + coefficients = { + (1, 0): RationalInterval.point(1), + (1, 1): RationalInterval.point(2), + (0, 1): RationalInterval.point(2), + } + certificate = certify_anticommuting_partition( + coefficients, (((1, 0), (1, 1), (0, 1)),) + ) + assert certificate.bound == 3 + + +def test_corrupt_commuting_group_is_rejected() -> None: + coefficients = { + (1, 0): RationalInterval.point(1), + (2, 0): RationalInterval.point(1), + } + with pytest.raises(ValueError, match="does not anticommute"): + certify_anticommuting_partition( + coefficients, (((1, 0), (2, 0)),) + ) +``` + +- [ ] **Step 2: Run the tests and verify import failure** + +Run: + +```bash +pytest -q tests/test_anticommuting.py +``` + +Expected: collection fails because `trottercert.anticommuting` does not exist. + +- [ ] **Step 3: Implement proof dataclasses and exact helpers** + +Create: + +```python +@dataclass(frozen=True) +class AnticommutingGroupCertificate: + term_indices: tuple[int, ...] + bound: Fraction + + +@dataclass(frozen=True) +class AnticommutingPartitionCertificate: + paulis: tuple[SymplecticPauli, ...] + coefficients: tuple[RationalInterval, ...] + groups: tuple[AnticommutingGroupCertificate, ...] + bound: Fraction +``` + +Implement `symplectic_anticommutes(left, right)` with the exact binary +symplectic parity expression. Implement an outward rational square-root +helper using `isqrt`, defaulting to 30 decimal digits and checking +`upper * upper >= value`. + +- [ ] **Step 4: Implement the deterministic discovery heuristic** + +Use: + +```python +def discover_anticommuting_partition( + coefficients: Mapping[SymplecticPauli, RationalInterval], + *, + max_group_size: int = 10, +) -> tuple[tuple[SymplecticPauli, ...], ...]: + if max_group_size < 1: + raise ValueError("maximum group size must be positive") + ordered = tuple( + sorted( + coefficients, + key=lambda pauli: ( + -float(coefficients[pauli].abs_upper()), + pauli, + ), + ) + ) + used: set[SymplecticPauli] = set() + groups: list[tuple[SymplecticPauli, ...]] = [] + for position, pauli in enumerate(ordered): + if pauli in used: + continue + group = [pauli] + used.add(pauli) + for candidate in ordered[position + 1:]: + if candidate in used: + continue + if all( + symplectic_anticommutes(candidate, member) + for member in group + ): + group.append(candidate) + used.add(candidate) + if len(group) == max_group_size: + break + groups.append(tuple(group)) + return tuple(groups) +``` + +Sort by `(-float(interval.abs_upper()), pauli)`. For every largest unused +Pauli, scan the remaining order and add a candidate exactly when it +anticommutes with every group member. Stop at `max_group_size`; preserve +singletons. + +- [ ] **Step 5: Implement independent exact certification** + +Use: + +```python +def certify_anticommuting_partition( + coefficients: Mapping[SymplecticPauli, RationalInterval], + groups: Sequence[Sequence[SymplecticPauli]], + *, + sqrt_decimal_places: int = 30, +) -> AnticommutingPartitionCertificate: + paulis = tuple(sorted(coefficients)) + index = {pauli: offset for offset, pauli in enumerate(paulis)} + flattened = tuple(pauli for group in groups for pauli in group) + if len(flattened) != len(set(flattened)): + raise ValueError("partition coverage contains duplicate terms") + if set(flattened) != set(paulis): + raise ValueError("partition coverage differs from coefficient map") + certified_groups = [] + for group in groups: + if not group: + raise ValueError("anticommuting group must be nonempty") + for left_position, left in enumerate(group): + for right in group[left_position + 1:]: + if not symplectic_anticommutes(left, right): + raise ValueError("group members do not anticommute") + squared = sum( + ( + coefficients[pauli].abs_upper() ** 2 + for pauli in group + ), + Fraction(), + ) + bound = sqrt_fraction_upper( + squared, decimal_places=sqrt_decimal_places + ) + certified_groups.append( + AnticommutingGroupCertificate( + tuple(index[pauli] for pauli in group), + bound, + ) + ) + return AnticommutingPartitionCertificate( + paulis, + tuple(coefficients[pauli] for pauli in paulis), + tuple(certified_groups), + sum((group.bound for group in certified_groups), Fraction()), + ) +``` + +Require exact set equality between coefficient keys and flattened groups, +reject duplicates, check every within-group pair, compute each squared +absolute upper sum, round its square root upward, and sum group bounds. + +- [ ] **Step 6: Add deterministic and coverage tests** + +Add: + +```python +def test_discovery_is_deterministic_and_covers_terms() -> None: + coefficients = { + (1, 0): RationalInterval.point(3), + (1, 1): RationalInterval.point(2), + (0, 1): RationalInterval.point(1), + } + first = discover_anticommuting_partition(coefficients, max_group_size=3) + second = discover_anticommuting_partition(coefficients, max_group_size=3) + assert first == second + certificate = certify_anticommuting_partition(coefficients, first) + assert set(certificate.paulis) == set(coefficients) +``` + +- [ ] **Step 7: Run and commit Task 2** + +Run: + +```bash +pytest -q tests/test_anticommuting.py +``` + +Then: + +```bash +git add tracks/qcs/solutions/WangTheoPhys/issue128/src/trottercert/anticommuting.py \ + tracks/qcs/solutions/WangTheoPhys/issue128/tests/test_anticommuting.py +git commit -m "certify anticommuting Pauli partitions" +``` + +--- + +### Task 3: Integrate the grouped D4 bound into resource evaluation + +**Files:** +- Modify: `src/trottercert/refined_error.py` +- Modify: `tests/test_refined_error.py` + +**Interfaces:** +- Consumes: `AnticommutingPartitionCertificate.bound`, a per-cell bound. +- Produces: optional `d4_site_override` in `evaluate_refined_fourth_order_bound`. + +- [ ] **Step 1: Write the failing override test** + +Add: + +```python +def test_d4_override_changes_only_degree_four() -> None: + constants = build_refined_fourth_order_constants() + original = evaluate_refined_fourth_order_bound(constants, 144, 116) + grouped = evaluate_refined_fourth_order_bound( + constants, + 144, + 116, + d4_site_override=constants.d4_site / 2, + ) + assert grouped.degree_four_contribution == ( + original.degree_four_contribution / 2 + ) + assert grouped.degree_five_contribution == original.degree_five_contribution + assert grouped.degree_six_contribution == original.degree_six_contribution + assert grouped.degree_seven_contribution == original.degree_seven_contribution + assert grouped.tail_contribution == original.tail_contribution +``` + +- [ ] **Step 2: Verify failure** + +Run: + +```bash +pytest -q tests/test_refined_error.py::test_d4_override_changes_only_degree_four +``` + +Expected: `TypeError` for unexpected keyword `d4_site_override`. + +- [ ] **Step 3: Implement the optional bound** + +Change the signature to: + +```python +def evaluate_refined_fourth_order_bound( + constants: RefinedFourthOrderConstants, + n_sites: int, + steps: int, + *, + d4_site_override: Fraction | None = None, +) -> RefinedFourthOrderBound: +``` + +Use `constants.d4_site` when the override is `None`. Reject a negative +override and reject an override larger than the original Pauli-l1 bound so +the grouped path cannot accidentally weaken the certificate. + +- [ ] **Step 4: Add the 98/97 discovery regression** + +Mark the expensive test slow. It regenerates coefficients, discovers and +certifies groups of size ten, evaluates steps 98 and 97, and asserts: + +```python +assert at_98.global_error_bound <= Fraction(1, 10**6) +assert at_97.global_error_bound > Fraction(1, 10**6) +assert certificate.bound < 4 * constants.d4_site +``` + +The last line compares the per-cell grouped D4 bound with the original +per-cell Pauli-l1 value. + +- [ ] **Step 5: Run focused tests and commit Task 3** + +Run: + +```bash +pytest -q tests/test_refined_error.py +pytest -q -m slow tests/test_refined_error.py::test_grouped_d4_crosses_fourfold_target +``` + +Commit: + +```bash +git add tracks/qcs/solutions/WangTheoPhys/issue128/src/trottercert/refined_error.py \ + tracks/qcs/solutions/WangTheoPhys/issue128/tests/test_refined_error.py +git commit -m "apply grouped D4 norm to refined bound" +``` + +--- + +### Task 4: Serialize and independently verify the large D4 proof + +**Files:** +- Modify: `src/trottercert/verify.py` +- Modify: `tests/test_resources_verify.py` +- Modify: `scripts/build_v3_certificate.py` +- Create: `certificates/issue128-d4-groups.json` + +**Interfaces:** +- Produces sidecar schema: + +```json +{ + "schema_version": 1, + "coefficient_denominator": 1, + "terms": [[0, 0, 0, 0]], + "sqrt_denominator": 1, + "groups": [[[0], 0]], + "cell_bound": [0, 1] +} +``` + +The real file uses a common exact coefficient denominator; each term is +`[x_mask, z_mask, lower_numerator, upper_numerator]`, and each group is +`[term_indices, bound_numerator]`. + +- [ ] **Step 1: Add failing fast-verifier corruption tests** + +Extend `test_resources_verify.py` to copy both certificate files to a +temporary directory and corrupt, one at a time: + +1. a repeated term index; +2. a within-group Pauli mask so the pair commutes; +3. a group-bound numerator reduced by one; +4. the sidecar hash in the main certificate. + +Each mutation must raise a `ValueError` containing a distinct stable phrase: +`coverage`, `anticommute`, `group bound`, or `sidecar digest`. + +- [ ] **Step 2: Add sidecar serialization to the builder** + +Add helper functions: + +```python +def d4_sidecar_payload( + certificate: AnticommutingPartitionCertificate, +) -> dict[str, object]: + coefficient_denominator = lcm( + *( + denominator + for interval in certificate.coefficients + for denominator in ( + interval.lower.denominator, + interval.upper.denominator, + ) + ) + ) + sqrt_denominator = lcm( + *(group.bound.denominator for group in certificate.groups) + ) + return { + "schema_version": 1, + "coefficient_denominator": coefficient_denominator, + "terms": [ + [ + pauli[0], + pauli[1], + interval.lower.numerator + * (coefficient_denominator // interval.lower.denominator), + interval.upper.numerator + * (coefficient_denominator // interval.upper.denominator), + ] + for pauli, interval in zip( + certificate.paulis, certificate.coefficients + ) + ], + "sqrt_denominator": sqrt_denominator, + "groups": [ + [ + list(group.term_indices), + group.bound.numerator + * (sqrt_denominator // group.bound.denominator), + ] + for group in certificate.groups + ], + "cell_bound": pair(certificate.bound), + } + + +def canonical_json_bytes(payload: dict[str, object]) -> bytes: + return ( + json.dumps(payload, sort_keys=True, separators=(",", ":")) + + "\n" + ).encode() +``` + +Write `issue128-d4-groups.json` from canonical bytes and store in the main +certificate: + +```python +"d4_certificate": { + "path": "issue128-d4-groups.json", + "sha256": hashlib.sha256(sidecar_bytes).hexdigest(), + "max_group_size": 10, + "cell_norm_upper": pair(partition.bound), +} +``` + +Search the minimum step count with +`d4_site_override=partition.bound / 4`. + +- [ ] **Step 3: Implement the fast sidecar verifier** + +Add: + +```python +def _verify_d4_sidecar( + certificate_path: Path, + candidate: dict[str, object], +) -> Fraction: + metadata = candidate["d4_certificate"] + root = certificate_path.resolve().parent + sidecar_path = (root / str(metadata["path"])).resolve() + if sidecar_path.parent != root: + raise ValueError("D4 sidecar path escapes certificate directory") + raw = sidecar_path.read_bytes() + if hashlib.sha256(raw).hexdigest() != metadata["sha256"]: + raise ValueError("D4 sidecar digest mismatch") + payload = json.loads(raw) + coefficient_denominator = int(payload["coefficient_denominator"]) + rows = payload["terms"] + paulis = tuple((int(row[0]), int(row[1])) for row in rows) + if len(paulis) != len(set(paulis)): + raise ValueError("D4 sidecar coefficient terms are duplicated") + coefficients = { + pauli: RationalInterval( + Fraction(int(row[2]), coefficient_denominator), + Fraction(int(row[3]), coefficient_denominator), + ) + for pauli, row in zip(paulis, rows) + } + submitted_groups = payload["groups"] + groups = tuple( + tuple(paulis[int(index)] for index in row[0]) + for row in submitted_groups + ) + regenerated = certify_anticommuting_partition(coefficients, groups) + sqrt_denominator = int(payload["sqrt_denominator"]) + submitted_bounds = tuple( + Fraction(int(row[1]), sqrt_denominator) + for row in submitted_groups + ) + regenerated_bounds = tuple( + group.bound for group in regenerated.groups + ) + if submitted_bounds != regenerated_bounds: + raise ValueError("D4 group bound mismatch") + cell_bound = _fraction(payload["cell_bound"]) + if cell_bound != regenerated.bound: + raise ValueError("D4 cell bound mismatch") + if cell_bound != _fraction(metadata["cell_norm_upper"]): + raise ValueError("D4 main-certificate bound mismatch") + return cell_bound / 4 +``` + +Resolve the sidecar relative to the main certificate, reject path traversal, +verify SHA-256, parse integer masks and common denominators, check exact +coverage and every pair, verify every squared rational bound, and require the +sum to equal `cell_bound`. Return the per-site value `cell_bound / 4`. + +- [ ] **Step 4: Wire fast and deep modes** + +Fast mode passes the returned site bound into +`evaluate_refined_fourth_order_bound`. Deep mode regenerates +`certified_d4_cell_coefficients`, reconstructs the ordered term table, +requires exact equality with the sidecar, and then invokes the same fast +proof checks. + +- [ ] **Step 5: Run corruption and fast-verification tests** + +Run: + +```bash +pytest -q tests/test_resources_verify.py +PYTHONPATH=src python scripts/build_v3_certificate.py +PYTHONPATH=src python scripts/verify.py \ + certificates/issue128-certificate.json +``` + +Expected certificate summary: + +```text +candidate_steps: 98 +candidate_group_exponentials: 2941 +global_twofold_target_met: true +improvement: greater than 4 +``` + +- [ ] **Step 6: Commit Task 4** + +```bash +git add tracks/qcs/solutions/WangTheoPhys/issue128/src/trottercert/verify.py \ + tracks/qcs/solutions/WangTheoPhys/issue128/tests/test_resources_verify.py \ + tracks/qcs/solutions/WangTheoPhys/issue128/scripts/build_v3_certificate.py \ + tracks/qcs/solutions/WangTheoPhys/issue128/certificates/issue128-d4-groups.json \ + tracks/qcs/solutions/WangTheoPhys/issue128/certificates/issue128-certificate.json +git commit -m "add machine-checkable grouped D4 certificate" +``` + +--- + +### Task 5: Rebuild cross-checks, transcripts, and documentation + +**Files:** +- Modify: `src/trottercert/crosscheck.py` +- Modify: `tests/test_small_crosscheck.py` +- Modify: `certificates/issue128-small-crosscheck.json` +- Modify: `certificates/issue128-verification.txt` +- Modify: `README.md` + +**Interfaces:** +- Consumes: the verified candidate step, error ledger, and exact resources. +- Produces: reproducible human-readable and dense-check artifacts. + +- [ ] **Step 1: Update the dense-check regression** + +Change `REFINED_STEPS` to the certificate step count and replace the +outward-rounded N=144 summary with the smallest decimal rational above the +exact grouped certificate total. Keep the N=4 scaling and assertions: + +```python +assert result["bound_dominates_empirical_error"] +assert result["bound_meets_tolerance"] +assert result["steps"] == 98 +``` + +- [ ] **Step 2: Regenerate the dense artifact** + +Run: + +```bash +PYTHONPATH=src python scripts/crosscheck_small.py \ + --length 2 --tolerance 1000000 +``` + +Copy the emitted values exactly into +`certificates/issue128-small-crosscheck.json`. + +- [ ] **Step 3: Run deep verification and capture the exact ledger** + +Run: + +```bash +PYTHONPATH=src python scripts/verify.py \ + certificates/issue128-certificate.json --deep +``` + +Update `issue128-verification.txt` with the emitted candidate error, +previous-step error, D4 grouped bound, steps, group count, bond count, CNOT +count, and exact improvement ratio. + +- [ ] **Step 4: Update README claims** + +Document: + +- the pairwise-anticommuting theorem; +- the fact that partition discovery is untrusted and proof checking is exact; +- the 98/97 integer boundary; +- `2941` candidate groups; +- `2941 * 72` bond propagators; +- three times that number for CNOTs; +- exact `11791/2941` improvement; +- updated test and deep-verification results. + +- [ ] **Step 5: Commit Task 5** + +```bash +git add tracks/qcs/solutions/WangTheoPhys/README.md \ + tracks/qcs/solutions/WangTheoPhys/issue128/src/trottercert/crosscheck.py \ + tracks/qcs/solutions/WangTheoPhys/issue128/tests/test_small_crosscheck.py \ + tracks/qcs/solutions/WangTheoPhys/issue128/certificates/issue128-small-crosscheck.json \ + tracks/qcs/solutions/WangTheoPhys/issue128/certificates/issue128-verification.txt +git commit -m "document fourfold certified resource reduction" +``` + +--- + +### Task 6: Full verification and PR update + +**Files:** +- Modify only if verification exposes an in-scope defect in files listed above. + +**Interfaces:** +- Produces: a clean pushed branch and updated PR #248. + +- [ ] **Step 1: Run the solution test suite** + +```bash +cd tracks/qcs/solutions/WangTheoPhys/issue128 +pytest -q +``` + +Expected: all default tests pass. + +- [ ] **Step 2: Run the deep proof regeneration** + +```bash +PYTHONPATH=src python scripts/verify.py \ + certificates/issue128-certificate.json --deep +``` + +Expected: `"valid": true`, `"verification_level": "deep"`, +`"deep_proof_regenerated": true`, candidate steps 98, and improvement above +four. + +- [ ] **Step 3: Run repository tests with the dependency-complete interpreter** + +From the repository root: + +```bash +python3.12 -m pytest scripts/tests/ -q \ + --cov=cluster_profile --cov=cluster_guardrail --cov=cluster_probe \ + --cov=parameter_scan --cov=scaling_fit \ + --cov-report=term-missing +``` + +Expected: 223 repository tests pass and measured coverage remains at least +95%. + +- [ ] **Step 4: Audit scope and formatting** + +```bash +git diff --check myfork/codex/issue-128-trotter-certificate...HEAD +git diff --name-only myfork/codex/issue-128-trotter-certificate...HEAD | + awk '!/^tracks\\/qcs\\/solutions\\/WangTheoPhys\\// {print}' +git status --short +``` + +Expected: no whitespace errors, no out-of-scope paths, and no uncommitted +changes. + +- [ ] **Step 5: Push the tested head to the PR branch** + +```bash +git push myfork HEAD:codex/issue-128-trotter-certificate +``` + +- [ ] **Step 6: Update and inspect PR #248** + +Set the title to the exact verified decimal rounded to three places: + +```bash +gh pr edit 248 --repo QuantumBFS/quantum.harness \ + --title "[qcs] 🔭 WangTheoPhys: certify a 4.009× tighter Trotter resource bound" \ + --body-file /tmp/issue128-pr-body.md +gh pr view 248 --repo QuantumBFS/quantum.harness \ + --json url,title,state,isDraft,headRefName,statusCheckRollup +``` + +The body must report only values copied from the verified certificate and +must retain the team name and member information already present in the +submission. + +--- + +## Self-Review + +- Spec coverage: Tasks 1-3 establish the mathematical bound; Task 4 makes it + independently machine-checkable; Task 5 updates every derived artifact; + Task 6 verifies and submits it. +- Trusted-computing-base boundary: discovery floats select groups only; + exact interval coefficients, anticommutation, square roots, coverage, + arithmetic, and hashing are rechecked. +- Type consistency: the coefficient map uses + `dict[SymplecticPauli, RationalInterval]`; discovery consumes that exact + type; certification returns `AnticommutingPartitionCertificate`; resource + evaluation consumes its `Fraction` cell bound divided by four. +- Scope consistency: all committed paths remain beneath the allowed + `WangTheoPhys` solution directory. diff --git a/tracks/qcs/solutions/WangTheoPhys/issue128/docs/superpowers/plans/2026-07-30-issue128-final-delivery.md b/tracks/qcs/solutions/WangTheoPhys/issue128/docs/superpowers/plans/2026-07-30-issue128-final-delivery.md new file mode 100644 index 000000000..1cb8c86e6 --- /dev/null +++ b/tracks/qcs/solutions/WangTheoPhys/issue128/docs/superpowers/plans/2026-07-30-issue128-final-delivery.md @@ -0,0 +1,188 @@ +# Issue 128 Final Delivery Implementation Plan + +> **For agentic workers:** REQUIRED SUB-SKILL: Use superpowers:subagent-driven-development (recommended) or superpowers:executing-plans to implement this plan task-by-task. Steps use checkbox (`- [ ]`) syntax for tracking. + +**Goal:** Publish a reviewer-ready Issue 128 package that freezes the certified 4.050498x result, adds exact data summaries and a polished PDF, and updates the existing PR without claiming an unproved fivefold certificate. + +**Architecture:** The committed certificate and exact sidecars remain the scientific source of truth. A small standard-library Python packager checks frozen hashes, projects exact fields into JSON/text, captures verifier evidence, and binds release files with SHA-256. LaTeX presents the same values for human review; remote publication happens only after an allowlisted audit. + +**Tech Stack:** Python 3.11+, pytest, existing `trottercert`, LaTeX/latexmk, BibTeX, Poppler, Git, GitHub CLI. + +## Global Constraints + +- Official result: 97 steps, 2,911 groups, exact ratio `11791/2911`. +- Baseline: 393 steps and 11,791 groups. +- The 97-step exact-rational error bound passes `1e-6`; the 96-step bound fails. +- Main certificate SHA-256: `0a09623ce3b292a3637065c870fb3153bbdcddce30aef968565c4db3ddfc7201`. +- D4 sidecar SHA-256: `a397414bb0229fb1ebdb38798aa781fb89dbb9d5cdbed94c7cd2e9120da62718`. +- D5 is follow-up evidence only; 78 steps and fivefold are not certified. +- Use standard `article` because no challenge template exists. +- Never touch the dirty main workspace, use `git add -A`, force-push, or create a competing PR. + +--- + +### Task 1: Restore and Validate Exact D5 Follow-up Evidence + +**Files:** +- Create: `src/trottercert/cubic_field.py` +- Create: `src/trottercert/cubic_local.py` +- Create: `src/trottercert/support_groups.py` +- Create: `scripts/build_d5_certificate.py` +- Create: `tests/test_cubic_field.py` +- Create: `tests/test_cubic_local.py` +- Create: `tests/test_support_groups.py` +- Generate: `certificates/issue128-d5-groups.json.gz` + +**Interfaces:** +- Consumes: the existing four-fragment Heisenberg symplectic evaluator and exact rational intervals. +- Produces: exact `Q(alpha)` Suzuki coefficients (`alpha^3=4`), an exact D5 Pauli density, deterministic same-support groups, canonical gzip bytes, and solver-free verification. + +- [ ] Run focused tests: + +```bash +pytest -q tests/test_cubic_field.py tests/test_support_groups.py +``` + +- [ ] Build with deterministic discovery and hard regression gates: + +```bash +env PYTHONHASHSEED=0 PYTHONPATH=src \ + python scripts/build_d5_certificate.py +``` + +Require 605,832 terms, 123,106 groups, exact site bound +`44948270001027856175670154896253/4000000000000000000000000000000`, and canonical SHA-256 +`c5e8968a93b4497b41fe42c0e364324388272e183e1fcd20a536bc988f5361dd`. + +- [ ] Verify from disk without discovery: + +```bash +PYTHONPATH=src python scripts/build_d5_certificate.py --verify-only +``` + +- [ ] Stage only the seven source/test files and the sidecar; commit as a single exact-follow-up unit. + +### Task 2: Deterministic Delivery Packager + +**Files:** +- Create: `scripts/package_delivery.py` +- Create: `scripts/__init__.py` +- Create: `tests/test_delivery_package.py` +- Generate: `artifacts/issue128-summary.json` +- Generate: `artifacts/issue128-summary.txt` +- Generate: `artifacts/verification-transcript.txt` + +**Interfaces:** +- Produces: `build_summary(root)`, `render_summary_text(summary)`, `capture_verification(root)`, `write_sha_manifest(root)`, and CLI modes `--manifest`/`--check`. + +- [ ] Write tests that require exact 393/97 steps, 11,791/2,911 groups, + frozen hashes, exact ratio, 97/96 boundary, D4/D5 counts, and + `fivefold_followup.status == "not_certified"`. +- [ ] Confirm the tests fail before the module exists. +- [ ] Implement exact invariant checks and canonical `sort_keys=True` JSON. +- [ ] Render a text summary containing `FIVEFOLD STATUS: NOT CERTIFIED` and + `No 78-step global error certificate is claimed or supplied.` +- [ ] Capture stdout/stderr and exit codes for: + +```bash +PYTHONPATH=src python scripts/verify.py certificates/issue128-certificate.json +PYTHONPATH=src python scripts/build_d5_certificate.py --verify-only +``` + +- [ ] Run `pytest -q tests/test_delivery_package.py`, then the full default suite. +- [ ] Explicitly stage and commit the packager and its tests. + +### Task 3: Technical Report Source + +**Files:** +- Create: `docs/report/issue128-technical-report.tex` +- Create: `docs/report/references.bib` +- Modify: `../README.md` (team README relative to the Issue 128 directory). + +**Interfaces:** +- Consumes: exact certificate/summary values and primary-source bibliography metadata. +- Produces: a self-contained 6--8 page report and reviewer navigation links. + +- [ ] Verify primary metadata for Childs--Su--Tran--Wiebe--Zhu and + Schubert--Mendl from their primary pages. +- [ ] Write the benchmark/result table with steps, groups, bond propagators, + CNOT upper bounds, exact ratio, and exact integer boundary. +- [ ] Explain the full-word logarithm, Dynkin--Specht--Wever projection, + concrete Pauli combination, D4 anticommuting certificate, local Heisenberg + lemma, right-generator ledger, tail, verifier, and 2x2 cross-check. +- [ ] Add a fivefold feasibility section that reports D5 evidence and names + the unresolved D4/D8 gates without any affirmative 5x claim. +- [ ] Add copy-paste reproduction commands and links to PDF/JSON/TXT/transcript/manifest. +- [ ] Commit report sources and README links explicitly. + +### Task 4: Build and Visually Audit the PDF + +**Files:** +- Generate: `docs/report/output/issue128-technical-report.pdf` + +- [ ] Generate summaries/transcript: + +```bash +PYTHONPATH=src python scripts/package_delivery.py +``` + +- [ ] Compile: + +```bash +cd docs/report +/Library/TeX/texbin/latexmk -pdf -interaction=nonstopmode \ + -halt-on-error -outdir=output issue128-technical-report.tex +``` + +- [ ] Reject fatal LaTeX errors, undefined references/citations, missing + glyphs, and visible overfull boxes. +- [ ] Require 6--8 nonblank pages and selectable text via `pdfinfo` and + `pdftotext`. +- [ ] Render every page at 144 DPI with `pdftoppm`; inspect each PNG for + clipping, overlap, broken tables, and inconsistent page furniture. +- [ ] Correct source and repeat compilation/rendering until the latest images + have zero visible defects. + +### Task 5: Bind Artifacts and Complete the Local Audit + +**Files:** +- Generate: `artifacts/SHA256SUMS` + +- [ ] Bind the PDF, summaries, transcript, main certificate, D4/D5 sidecars, + small cross-check, and report sources. +- [ ] Run `shasum -a 256 -c artifacts/SHA256SUMS` and require every line `OK`. +- [ ] Run: + +```bash +pytest -q +PYTHONPATH=src python scripts/verify.py certificates/issue128-certificate.json +PYTHONPATH=src python scripts/build_d5_certificate.py --verify-only +PYTHONPATH=src python scripts/package_delivery.py --check +``` + +- [ ] Scan for unsupported fivefold language and placeholder/tool tokens. +- [ ] Run `git diff --check c550a6b..HEAD` and audit every changed path. +- [ ] Require a clean worktree after explicitly staging/committing final artifacts. + +### Task 6: Fast-forward the Existing PR + +**Remote target:** `JunkaiWang-TheoPhy:codex/issue-128-trotter-certificate` + +- [ ] Fetch the fork branch and re-read PR #248 head SHA/state. +- [ ] Require `git merge-base --is-ancestor origin/codex/issue-128-trotter-certificate HEAD`. +- [ ] Push `HEAD:codex/issue-128-trotter-certificate` without force. +- [ ] Update PR #248 body with the exact result table, innovation list, + artifact links, commands, test/verifier results, and visible fivefold non-claim. +- [ ] Fetch again and require remote SHA equals audited local SHA. +- [ ] Report the PDF/data paths, final commit, PR URL, validation evidence, + achieved 4.050498x ratio, and external maintainer review/merge condition. + +## Self-Review + +- Every requested deliverable maps to Tasks 1--6. +- Every numerical claim is either an exact certificate field or explicitly + marked conditional follow-up arithmetic. +- The PDF is not accepted based on compilation alone; every page is rendered + and inspected. +- The remote write is downstream of all proof, artifact, visual, hash, diff, + and drift gates. diff --git a/tracks/qcs/solutions/WangTheoPhys/issue128/docs/superpowers/plans/2026-07-31-issue128-six-hour-hpc-sprint.md b/tracks/qcs/solutions/WangTheoPhys/issue128/docs/superpowers/plans/2026-07-31-issue128-six-hour-hpc-sprint.md new file mode 100644 index 000000000..29f0c3172 --- /dev/null +++ b/tracks/qcs/solutions/WangTheoPhys/issue128/docs/superpowers/plans/2026-07-31-issue128-six-hour-hpc-sprint.md @@ -0,0 +1,334 @@ +# Issue 128 Six-Hour HPC Sprint Implementation Plan + +> **For agentic workers:** REQUIRED SUB-SKILL: Use superpowers:executing-plans to implement this plan task-by-task. Steps use checkbox (`- [ ]`) syntax for tracking. + +**Goal:** Build, deploy, monitor, and recover a deterministic Slurm sprint that regenerates the accepted Issue 128 proof, computes an exact sharded degree-eight right-generator coefficient map, and screens all 24 matching-color orderings without weakening the frozen `4.050498110614909x` certificate. + +**Architecture:** The trusted branch remains anchored at `ebb055926302c7d911e5ab00805513c382ac458d`. Exact D8 work is decomposed into one independent contribution per one of the 31 merged Suzuki stages; every shard serializes coordinate-addressed Pauli coefficients in `Q[cuberoot(4)]`, so processes with different bit registries can be reduced safely. A reducer verifies all manifests and hashes, sums exact coefficients before any norm inequality, and emits a rigorous Pauli-l1 site-density upper bound. A separate untrusted discovery array evaluates all 24 fragment permutations with the existing floating-point right-generator recurrence. + +**Tech Stack:** Python 3.12, `Fraction`, exact `Cubic`, gzip JSON, pytest, Bash, Slurm arrays, `scripts/harness_slurm.sh`, SSH/rsync. + +## Global Constraints + +- Physical benchmark is fixed: periodic square spin-1/2 isotropic Heisenberg, `L=12`, `N=144`, `T=1`, operator-norm tolerance `1e-6`. +- Frozen accepted claim is `11791/2911 = 4.050498110614909...`; no new ratio may be claimed unless a complete exact global ledger and fast/deep verification pass. +- D8 shards must be merged coefficient-by-coefficient before taking any norm; shard norms must never be triangle-summed. +- Discovery may use float ranking, but trusted outputs use integers, `Fraction`, `Cubic`, outward rational intervals, and SHA-256. +- Each Slurm cell uses one CPU thread, `PYTHONHASHSEED=0`, and BLAS thread counts fixed to one. +- No computation runs on a login node. All remote compute is submitted to `xhacnormalb` under account `giggleliu` and the user's assigned QOS. +- Every cell writes an atomic manifest with commit, arguments, status, timing, counts, and output digest. +- The sprint stops launching discovery jobs after 4 hours 15 minutes and reserves the final 1 hour 45 minutes for reduction, verification, fetch, and packaging. + +--- + +### Task 1: Exact stage-contribution recurrence + +**Files:** +- Modify: `src/trottercert/cubic_local.py` +- Test: `tests/test_cubic_right_generator.py` + +**Interfaces:** +- Consumes: `Cubic`, `CubicStage`, `fourth_order_suzuki_cubic_stages`, `cubic_fragment_adjoint`. +- Produces: `exact_right_generator_stage_contribution(stages: Sequence[CubicStage], stage_index: int, order: int) -> tuple[CoordinateRegistry, list[CubicTerms]]` and `merge_cubic_series(target: list[CubicTerms], source: Sequence[Mapping[SymplecticPauli, Cubic]]) -> None`. + +- [ ] **Step 1: Write failing algebra tests** + +```python +def test_stage_contributions_reconstruct_order_conditions() -> None: + stages = fourth_order_suzuki_cubic_stages() + shards = [exact_right_generator_stage_contribution(stages, i, 4) for i in range(len(stages))] + coordinate_series = merge_coordinate_series([coordinate_encode_series(*shard) for shard in shards]) + assert not coordinate_series[1] + assert not coordinate_series[2] + assert not coordinate_series[3] + assert coordinate_series[4] + +def test_rejects_invalid_stage_index() -> None: + with pytest.raises(IndexError): + exact_right_generator_stage_contribution(fourth_order_suzuki_cubic_stages(), 31, 8) +``` + +- [ ] **Step 2: Run the focused tests and observe missing interfaces** + +Run: `pytest -q tests/test_cubic_right_generator.py` + +Expected: import failure for `exact_right_generator_stage_contribution`. + +- [ ] **Step 3: Implement exact conjugation and stage contributions** + +```python +def exact_right_generator_stage_contribution(stages, stage_index, order): + stage = stages[stage_index] + registry, base = exact_matching_density(stage.fragment_index) + series = [{} for _ in range(order + 1)] + series[0] = {pauli: stage.coefficient * coefficient for pauli, coefficient in base.items()} + for later in stages[stage_index + 1:]: + series = conjugate_cubic_series_by_stage(registry, series, later.fragment_index, later.coefficient) + return registry, [canonicalize_cubic_density(registry, degree) for degree in series] +``` + +- [ ] **Step 4: Verify exact cancellation through degree three** + +Run: `pytest -q tests/test_cubic_right_generator.py` + +Expected: all tests pass and degree four is nonempty. + +--- + +### Task 2: Registry-independent serialization and atomic manifests + +**Files:** +- Create: `src/trottercert/hpc_artifacts.py` +- Test: `tests/test_hpc_artifacts.py` + +**Interfaces:** +- Consumes: `CoordinateRegistry`, `CubicTerms`. +- Produces: `coordinate_encode_series`, `write_shard_gzip`, `read_shard_gzip`, `write_manifest_atomic`, and `sha256_file`. + +- [ ] **Step 1: Write round-trip and corruption tests** + +```python +def test_coordinate_round_trip_is_registry_independent(tmp_path: Path) -> None: + first_registry, first = build_fixture_with_registration_order("xy") + second_registry, second = build_fixture_with_registration_order("yx") + assert coordinate_encode_terms(first_registry, first) == coordinate_encode_terms(second_registry, second) + +def test_reader_rejects_payload_digest_mismatch(tmp_path: Path) -> None: + path = tmp_path / "shard.json.gz" + write_shard_gzip(path, fixture_payload()) + raw = bytearray(path.read_bytes()) + raw[-8] ^= 1 + path.write_bytes(raw) + with pytest.raises((OSError, ValueError)): + read_shard_gzip(path) +``` + +- [ ] **Step 2: Run tests and confirm missing functions** + +Run: `pytest -q tests/test_hpc_artifacts.py` + +- [ ] **Step 3: Implement canonical coordinate keys and fraction triples** + +```python +def encode_cubic(value: Cubic) -> list[list[int]]: + return [[part.numerator, part.denominator] for part in (value.a0, value.a1, value.a2)] + +def coordinate_pauli_key(registry, pauli): + return tuple(sorted((x, y, op) for x, y, op in pauli_coordinates(registry, pauli))) +``` + +Use sorted keys, compact JSON separators, gzip `mtime=0`, temporary sibling files, `fsync`, and `Path.replace`. + +- [ ] **Step 4: Run deterministic double-write test** + +Run: `pytest -q tests/test_hpc_artifacts.py` + +Expected: two writes of the same payload have identical SHA-256. + +--- + +### Task 3: D8 shard CLI and reducer + +**Files:** +- Create: `scripts/build_d8_shard.py` +- Create: `scripts/reduce_d8_shards.py` +- Test: `tests/test_d8_cli.py` + +**Interfaces:** +- `build_d8_shard.py --stage-index I --order 8 --output PATH --manifest PATH` writes one exact shard. +- `reduce_d8_shards.py --input-root DIR --expected-stages 31 --output PATH --summary PATH` verifies and reduces all shards. + +- [ ] **Step 1: Write CLI tests using order four fixtures** + +```python +def test_all_stage_shards_reduce_to_exact_order_conditions(tmp_path: Path) -> None: + for index in range(31): + run_shard(index=index, order=4, root=tmp_path) + summary = reduce_shards(tmp_path, expected_stages=31, order=4) + assert summary["missing_stages"] == [] + assert summary["degree_term_counts"]["1"] == 0 + assert summary["degree_term_counts"]["2"] == 0 + assert summary["degree_term_counts"]["3"] == 0 + assert summary["degree_term_counts"]["4"] > 0 +``` + +- [ ] **Step 2: Implement the shard CLI with progress and failure manifests** + +The CLI writes `status="running"` before compute, prints a flushed progress line, writes the payload, then atomically replaces the manifest with `status="complete"`. Exceptions atomically write `status="failed"` and are re-raised. + +- [ ] **Step 3: Implement exact reduction** + +Reducer checks stage set `0..30`, order, formula id, git commit, payload SHA, and unique stage indices. It sums coordinate-addressed `Cubic` coefficients exactly. For degree eight, it encloses every coefficient with the registered cuberoot interval and sums `max(abs(lower), abs(upper))` into an exact rational Pauli-l1 site-density upper bound. + +- [ ] **Step 4: Run CLI tests** + +Run: `pytest -q tests/test_d8_cli.py` + +--- + +### Task 4: Matching-order discovery array + +**Files:** +- Create: `scripts/screen_matching_order.py` +- Test: `tests/test_matching_order_screen.py` + +**Interfaces:** +- `screen_matching_order.py --permutation-index I --order 6 --output PATH --manifest PATH` evaluates one of `itertools.permutations(range(4))`. + +- [ ] **Step 1: Write permutation coverage tests** + +```python +def test_permutation_index_covers_exactly_24_orders() -> None: + values = [permutation_from_index(i) for i in range(24)] + assert len(set(values)) == 24 + assert values[0] == (0, 1, 2, 3) +``` + +- [ ] **Step 2: Implement float discovery only** + +Remap each `ScalarStage.fragment_index`, run `right_generator_local_series(..., order=6)`, and record term count plus coefficient-l1 for degrees four through six. The manifest must say `trusted=false` and `purpose="discovery-ranking"`. + +- [ ] **Step 3: Verify deterministic identity result** + +Run: `pytest -q tests/test_matching_order_screen.py` + +--- + +### Task 5: Slurm profiles, run specs, and sbatch scripts + +**Files:** +- Create: `skills/using-slurm/profiles/xh5-acamtw.toml` +- Create: `skills/using-slurm/profiles/xh5-cfys.toml` +- Create: `hpc/issue128_d8_array.sbatch` +- Create: `hpc/issue128_d8_reduce.sbatch` +- Create: `hpc/issue128_screen_array.sbatch` +- Create: `hpc/issue128_verify.sbatch` +- Create: `scripts/plan_issue128_hpc.py` +- Test: `tests/test_issue128_hpc_plan.py` + +**Interfaces:** +- Profiles select `xhacnormalb`, account `giggleliu`, and the matching user QOS. +- Planner writes `results//run_spec.json` with 31 D8 cells and 24 ordering cells. + +- [ ] **Step 1: Write plan-schema tests** + +```python +def test_sprint_plan_has_unique_cells_and_bounded_resources(tmp_path: Path) -> None: + spec = build_run_spec("issue128-hpc-test") + ids = [cell["cell_id"] for cell in spec["cells"]] + assert len(ids) == len(set(ids)) == 55 + assert sum(cell["kind"] == "d8-stage" for cell in spec["cells"]) == 31 + assert sum(cell["kind"] == "ordering-screen" for cell in spec["cells"]) == 24 +``` + +- [ ] **Step 2: Add fail-closed profiles and scripts** + +Profiles set hard ceilings of six hours, one node, 64 CPUs, and 200 array cells. Sbatch scripts request one CPU per cell, fixed thread environment, explicit memory, array concurrency caps, and no GPU GRES. + +- [ ] **Step 3: Inspect guardrails and dry-run submission** + +Run: + +```bash +python3 scripts/cluster_guardrail.py inspect hpc/issue128_d8_array.sbatch --profile skills/using-slurm/profiles/xh5-acamtw.toml +scripts/harness_slurm.sh --profile skills/using-slurm/profiles/xh5-acamtw.toml --dry-run submit --script hpc/issue128_d8_array.sbatch +``` + +Expected: no hard-limit violation and no remote command executed. + +--- + +### Task 6: Local proof-preservation gate + +**Files:** +- Modify: `tests/test_delivery_package.py` + +- [ ] **Step 1: Run focused new tests** + +Run: `pytest -q tests/test_cubic_right_generator.py tests/test_hpc_artifacts.py tests/test_d8_cli.py tests/test_matching_order_screen.py tests/test_issue128_hpc_plan.py` + +- [ ] **Step 2: Run accepted-certificate regression** + +Run: + +```bash +PYTHONPATH=src python scripts/verify.py certificates/issue128-certificate.json +PYTHONPATH=src python scripts/package_delivery.py --check +shasum -a 256 -c artifacts/SHA256SUMS +``` + +Expected: `valid=true`, exact ratio `11791/2911`, delivery PASS, and all ten hashes OK. + +- [ ] **Step 3: Run the complete default suite** + +Run: `pytest -q` + +Expected: no regressions. + +--- + +### Task 7: Remote bootstrap, smoke, and production submission + +**Files:** +- Runtime outputs only: `results//job-record.json`, cell logs, manifests, sidecars. + +- [ ] **Step 1: Precheck and probe** + +```bash +scripts/harness_slurm.sh --profile skills/using-slurm/profiles/xh5-acamtw.toml precheck +scripts/harness_slurm.sh --profile skills/using-slurm/profiles/xh5-acamtw.toml probe-partitions +``` + +- [ ] **Step 2: Ship exact source** + +Rsync only the clean sprint checkout, excluding `.git`, existing results, caches, and virtual environments. Record local commit and post-rsync remote file-list digest in `job-record.json`. + +- [ ] **Step 3: Bootstrap once on the login node** + +Create `.venv`, install `-e '.[test]'`, and run import-only smoke. Do not execute D8 on the login node. + +- [ ] **Step 4: Submit four-cell smoke** + +Submit D8 stages `0-3` with `--array=0-3%4`, wait for RUNNING, inspect one startup log, fetch four manifests, and require all four `status=complete` before production. + +- [ ] **Step 5: Submit production arrays and dependent reducer** + +Submit 31 D8 stages with concurrency 16, 24 ordering screens with concurrency 12, deep verifier, and full tests. Increase D8 concurrency to 32 only if smoke RSS and queue reasons allow. The reducer uses `afterok:`. + +--- + +### Task 8: Monitor, fetch, reduce, and adjudicate the 5x gate + +**Files:** +- Runtime outputs: `results//` +- Modify after evidence exists: `artifacts/verification-transcript.txt`, delivery summary, and report source. + +- [ ] **Step 1: Monitor scheduler and one representative log** + +Check PENDING→RUNNING, pending reasons, startup output, and 30–45 minute pulses. Do not treat scheduler completion as proof success. + +- [ ] **Step 2: Fetch and classify** + +```bash +scripts/harness_slurm.sh --profile skills/using-slurm/profiles/xh5-acamtw.toml fetch +scripts/harness_slurm.sh --profile skills/using-slurm/profiles/xh5-acamtw.toml classify +``` + +- [ ] **Step 3: Verify every cell and the reducer** + +Require 31/31 D8 manifests, digest matches, exact merge reproducibility under forward and reverse shard order, and a reducer summary with the exact D8 site-density bound. + +- [ ] **Step 4: Evaluate the global claim boundary** + +Only if an exact r=78 ledger is at most `1e-6`, the previous integer step fails, and fast/deep/fresh-clone verification pass may a 5x claim be added. Otherwise preserve `4.050498x` and label all new results as feasibility or negative evidence. + +- [ ] **Step 5: Final package audit** + +Render the PDF, inspect every page, regenerate JSON/TXT/transcript/SHA manifest, and reproduce from a fresh checkout before declaring the sprint complete. + +## Self-Review + +- Spec coverage: exact D8, discovery array, both Slurm accounts, six-hour ceiling, manifests, proof preservation, smoke/production/fetch, and final claim gate are assigned to Tasks 1–8. +- Placeholder scan: the plan contains no `TBD`, `TODO`, deferred implementation, or undefined “appropriate handling” steps. +- Type consistency: all D8 paths use `CubicTerms`; cross-process interchange uses coordinate-addressed keys; reducers never merge raw registry bit masks. +- Safety: GPU partitions are excluded, login-node compute is excluded, array concurrency is bounded, and the existing certificate remains authoritative unless the complete new gate passes. diff --git a/tracks/qcs/solutions/WangTheoPhys/issue128/docs/superpowers/plans/2026-07-31-proof-carrying-trotter-paper.md b/tracks/qcs/solutions/WangTheoPhys/issue128/docs/superpowers/plans/2026-07-31-proof-carrying-trotter-paper.md new file mode 100644 index 000000000..a057b7a32 --- /dev/null +++ b/tracks/qcs/solutions/WangTheoPhys/issue128/docs/superpowers/plans/2026-07-31-proof-carrying-trotter-paper.md @@ -0,0 +1,426 @@ +# Proof-Carrying Trotter Bound Paper Implementation Plan + +> **For agentic workers:** REQUIRED SUB-SKILL: Use superpowers:subagent-driven-development (recommended) or superpowers:executing-plans to implement this plan task-by-task. Steps use checkbox (`- [ ]`) syntax for tracking. + +**Goal:** Produce a verified 22--30 page English research manuscript that presents the Issue 128 result as a proof-carrying, model-aware compiler for rigorous Trotter resource bounds. + +**Architecture:** Keep the frozen certificate package immutable and build a separate modular LaTeX project under `docs/manuscript/`. Each scientific section consumes committed evidence or primary literature, while deterministic scripts generate figures and validate copied numerical claims. Compile with `latexmk`, then render every PDF page with Poppler for visual review. + +**Tech Stack:** LaTeX, BibTeX, latexmk, Python 3.12, matplotlib, JSON, gzip, exact committed certificate artifacts, Poppler. + +## Global Constraints + +- Preserve all existing uncommitted HPC work and do not stage it with manuscript commits. +- Do not modify the manifest-bound competition report or certificate artifacts. +- The only certified global result is `r=97`, 2,911 groups, and exact ratio `11791/2911`. +- State explicitly that no `r=78` global certificate is claimed or supplied. +- Use primary-source bibliographic metadata; do not invent references. +- Keep the manuscript in English and target at least 15 substantive PDF pages. +- Separate certified claims, empirical cross-checks, and conditional research directions. +- Use exact values from committed artifacts; plotted decimal values are display-only. + +--- + +## File map + +- `docs/manuscript/main.tex`: document class, packages, metadata, section order. +- `docs/manuscript/references.bib`: verified primary-source bibliography. +- `docs/manuscript/sections/abstract.tex`: scoped abstract and keywords. +- `docs/manuscript/sections/introduction.tex`: motivation, gap, contributions. +- `docs/manuscript/sections/problem.tex`: Hamiltonian, formula, norm, cost model. +- `docs/manuscript/sections/baseline.tex`: PRX theorem versus pinned instantiation. +- `docs/manuscript/sections/compiler.tex`: compiler data flow and invariants. +- `docs/manuscript/sections/free_lie.tex`: free-word logarithm and Lie projection. +- `docs/manuscript/sections/pauli.tex`: concrete symplectic Pauli evaluation. +- `docs/manuscript/sections/norm_certificate.tex`: canonicalization and anticommuting groups. +- `docs/manuscript/sections/finite_step.tex`: right generator and rational tail. +- `docs/manuscript/sections/verification.tex`: proof-carrying verifier architecture. +- `docs/manuscript/sections/results.tex`: resource result, ledgers, ablations. +- `docs/manuscript/sections/related_work.tex`: 2021--2026 literature taxonomy. +- `docs/manuscript/sections/limitations.tex`: fivefold and tenfold frontier. +- `docs/manuscript/sections/reproducibility.tex`: commands and evidence matrix. +- `docs/manuscript/sections/conclusion.tex`: scoped conclusion. +- `docs/manuscript/sections/appendices.tex`: technical derivations and schemas. +- `docs/manuscript/scripts/generate_figures.py`: deterministic PDF figure generation. +- `docs/manuscript/scripts/validate_claims.py`: compare manuscript constants with artifacts. +- `docs/manuscript/figures/*.pdf`: generated vector figures. +- `docs/manuscript/output/pdf/issue128-proof-carrying-trotter-paper.pdf`: final PDF. + +### Task 1: Initialize the manuscript project and evidence validator + +**Files:** +- Create: `docs/manuscript/main.tex` +- Create: `docs/manuscript/sections/*.tex` +- Create: `docs/manuscript/scripts/validate_claims.py` +- Create: `docs/manuscript/output/pdf/.gitkeep` + +**Interfaces:** +- Consumes: committed `artifacts/issue128-summary.json` and `certificates/issue128-certificate.json`. +- Produces: a compilable manuscript skeleton and `validate_claims.py` returning exit code zero only when all headline constants match. + +- [ ] **Step 1: Scan the existing LaTeX structure** + +Run: + +```bash +python3 /Users/thomasjwang/.codex/skills/academic-writer/scripts/writer_tools.py scan_template docs/report +``` + +Expected: identify the existing generic `article` report and its bibliography setup. + +- [ ] **Step 2: Create the modular manuscript skeleton** + +Use `article` at 11 pt with `geometry`, `amsmath`, `amssymb`, `mathtools`, `booktabs`, `microtype`, `graphicx`, `xcolor`, `hyperref`, `cleveref`, `algorithm`, `algpseudocode`, `siunitx`, and `natbib`. Set the section order exactly as listed in the file map and place final output under `output/pdf/`. + +- [ ] **Step 3: Implement exact headline-claim validation** + +`validate_claims.py` must load the summary and certificate, assert: + +```python +assert summary["published_baseline"]["steps"] == 393 +assert summary["certified_result"]["steps"] == 97 +assert summary["published_baseline"]["group_exponentials"] == 11791 +assert summary["certified_result"]["group_exponentials"] == 2911 +assert Fraction(11791, 2911) > 4 +``` + +It must also assert the accepted and rejected error fractions against `10**-6` using integer cross multiplication. + +- [ ] **Step 4: Run the validator** + +Run: + +```bash +python docs/manuscript/scripts/validate_claims.py +``` + +Expected output: + +```text +headline_claims=PASS +``` + +- [ ] **Step 5: Compile the skeleton** + +Run: + +```bash +cd docs/manuscript +latexmk -pdf -interaction=nonstopmode -halt-on-error main.tex +``` + +Expected: successful one-page or two-page structural PDF with no LaTeX errors. + +### Task 2: Build the verified bibliography and related-work taxonomy + +**Files:** +- Create: `docs/manuscript/references.bib` +- Create: `docs/manuscript/sections/related_work.tex` + +**Interfaces:** +- Consumes: primary pages for each cited paper. +- Produces: citation keys used throughout the manuscript and a comparison taxonomy that distinguishes direct bounds from changed-task algorithms. + +- [ ] **Step 1: Add verified baseline and model-specific references** + +Include complete BibTeX for Childs et al. PRX 2021 and Schubert--Mendl 2023, with DOI or arXiv identifiers checked against the primary publisher or arXiv record. + +- [ ] **Step 2: Add special-state and average-case references** + +Include the low-energy subspace paper, Chen--Brandao average-case paper, strong state-dependent error bounds, and Trotter lower bounds. + +- [ ] **Step 3: Add changed-algorithm references** + +Include dynamic MPF, explicit MPF commutator scaling, randomized order doubling, Chebyshev interpolation, Trotter error mitigation, Trotter LCU, THRIFT, optimized Suzuki schemes, commutation-based ordering, practical BCH estimation, and HNCC. + +- [ ] **Step 4: Write the taxonomy** + +Use four subsections: + +1. worst-case commutator-scaling upper bounds; +2. low-energy, average-case, and observable-specific bounds; +3. MPF, LCU, interpolation, randomization, and compensation; +4. formula and ordering optimization. + +For every comparison, state whether the work changes the circuit, state set, norm, sampling cost, or Hamiltonian model. + +- [ ] **Step 5: Compile and reject unresolved citations** + +Run: + +```bash +cd docs/manuscript +latexmk -pdf -interaction=nonstopmode -halt-on-error main.tex +rg -n "undefined citations|Citation.*undefined|There were undefined" main.log +``` + +Expected: `rg` returns no matches. + +### Task 3: Write the problem, baseline, and contribution boundary + +**Files:** +- Modify: `docs/manuscript/sections/abstract.tex` +- Modify: `docs/manuscript/sections/introduction.tex` +- Modify: `docs/manuscript/sections/problem.tex` +- Modify: `docs/manuscript/sections/baseline.tex` + +**Interfaces:** +- Consumes: frozen benchmark values and bibliography keys. +- Produces: a self-contained statement of the scientific question and a non-misleading baseline reconstruction. + +- [ ] **Step 1: Write the abstract** + +Include the exact benchmark, unchanged circuit family, `393 -> 97`, `11791/2911`, 75.3117 percent reduction, machine certificate, and explicit non-claim for fivefold. + +- [ ] **Step 2: Write the introduction** + +Motivate the gap between general asymptotic bounds and finite-instance resource estimates. End with a numbered list of five contributions: multi-IR compiler, norm-last Pauli evaluation, certified anticommuting partitions, finite-step rational closure, and independent artifact verification. + +- [ ] **Step 3: Define the problem** + +Define `H`, four matchings, normalized bond, `S_4`, global operator norm, `T=1`, `epsilon=10^-6`, `G(r)=30r+1`, 72 bonds per group, and three-CNOT upper bound. + +- [ ] **Step 4: Reconstruct the baseline carefully** + +Use the phrase "pinned interval instantiation of a published theorem." Explain the 31 admissible centers, center 20, Pauli-l1 procedure, site-density bound, and `r=393`. Do not imply that PRX 2021 printed the number 393. + +- [ ] **Step 5: Compile and inspect extracted text** + +Run: + +```bash +cd docs/manuscript +latexmk -pdf -interaction=nonstopmode -halt-on-error main.tex +pdftotext main.pdf - | sed -n '1,220p' +``` + +Expected: abstract and claim boundary read coherently without missing symbols. + +### Task 4: Write the compiler and algebraic method + +**Files:** +- Modify: `docs/manuscript/sections/compiler.tex` +- Modify: `docs/manuscript/sections/free_lie.tex` +- Modify: `docs/manuscript/sections/pauli.tex` +- Modify: `docs/manuscript/sections/norm_certificate.tex` + +**Interfaces:** +- Consumes: implementation modules in `src/trottercert/` and committed D4/D5 statistics. +- Produces: precise algorithms, invariants, and verifier obligations for every structural optimization. + +- [ ] **Step 1: Specify compiler inputs, IRs, and outputs** + +Define source input `(H, F, S, T, epsilon, C)`, free-word IR, Lie IR, Pauli IR, certificate IR, and output `(r_star, resources, certificate)`. + +- [ ] **Step 2: Add compiler pseudocode** + +The pseudocode must include exact coefficient parsing, free-word multiplication, formal logarithm, order verification, DSW projection, local Pauli evaluation, canonicalization, identical-term aggregation, grouping, finite-tail closure, adjacent-step search, and resource compilation. + +- [ ] **Step 3: Explain free-word formal logarithms** + +Define truncated sparse series, convolution, `log(1+X)`, symmetry cancellations, degree-five and degree-seven defects, and why no matrix of dimension `2^144` is formed. + +- [ ] **Step 4: Explain DSW projection** + +Give the right-nested commutator map, homogeneous-degree factor, and the invariant checked by the implementation. + +- [ ] **Step 5: Explain concrete Pauli evaluation** + +Define symplectic `(x,z)` masks, phase tracking, commutation parity, local-support filtering, translation representatives, and exact coefficient aggregation. + +- [ ] **Step 6: Explain anticommuting certification** + +Prove that pairwise anticommutation gives an l2 group norm. Describe the graph partition search as untrusted and list exact coverage, pairwise relation, coefficient, square-root, and hash checks. + +- [ ] **Step 7: Quantify the D4 improvement** + +Report 75,324 terms, 7,576 groups, maximum size 10, direct cell bound `20.160968407335066`, grouped bound `6.472926505087888`, and factor `3.114660484942633`. + +### Task 5: Write finite-step analysis, soundness, and resource compilation + +**Files:** +- Modify: `docs/manuscript/sections/finite_step.tex` +- Modify: `docs/manuscript/sections/verification.tex` +- Modify: `docs/manuscript/sections/appendices.tex` + +**Interfaces:** +- Consumes: right-generator identity, local lemma, error ledger, verifier behavior. +- Produces: a theorem-level argument connecting local algebra to the global resource result. + +- [ ] **Step 1: Prove the Heisenberg local lemma** + +Enumerate the 16 phase-free two-qubit Paulis in an appendix table and prove that each anticommutes with zero or exactly two of `XX`, `YY`, `ZZ`, yielding bond Pauli-l1 growth constant one. + +- [ ] **Step 2: Derive the right logarithmic generator through degree seven** + +Present the `D4`, `D5`, `D6`, and `D7` terms with their exact coefficients and explain the degree shift from log defects to the right generator. + +- [ ] **Step 3: State the finite-step soundness theorem** + +The theorem must condition on verified free-word coefficients, Pauli ledgers, group partitions, local growth lemma, and tail ratio. Its conclusion must be the certified global operator-norm upper bound for integer `r`. + +- [ ] **Step 4: Explain the rational tail** + +Describe support growth, conjugation Taylor factorials, Duhamel integration, the geometric ratio condition, and outward rational arithmetic. State that omitted degrees are never silently discarded. + +- [ ] **Step 5: Derive resource arithmetic** + +Prove `G(r)=30r+1`, `B(r)=72G(r)`, and `CNOT(r)<=3B(r)`. Cross-multiply exact fractions for `r=97` and `r=96`. + +- [ ] **Step 6: Specify the trusted computing base** + +Separate discovery code from trusted verification. List integers, `Fraction`, outward rational intervals, symplectic multiplication, digest verification, negative mutation tests, and deep regeneration. + +### Task 6: Generate figures and write results, ablations, and limitations + +**Files:** +- Create: `docs/manuscript/scripts/generate_figures.py` +- Create: `docs/manuscript/figures/resources.pdf` +- Create: `docs/manuscript/figures/error_ledger.pdf` +- Create: `docs/manuscript/figures/d4_norm.pdf` +- Create: `docs/manuscript/figures/fivefold_gap.pdf` +- Modify: `docs/manuscript/sections/results.tex` +- Modify: `docs/manuscript/sections/limitations.tex` + +**Interfaces:** +- Consumes: frozen summary and committed fivefold audit. +- Produces: deterministic vector figures and a results narrative with explicit certified/conditional labels. + +- [ ] **Step 1: Implement deterministic figure generation** + +Use matplotlib with embedded data loaded from JSON where available. Use a fixed style, vector PDF output, color-blind-safe colors, and labels that remain legible at single-column width. + +- [ ] **Step 2: Generate the resource comparison** + +Plot baseline and candidate values for steps, groups, bond propagators, and CNOTs using normalized or log-scaled panels so all metrics remain readable. + +- [ ] **Step 3: Generate the `r=97` error ledger** + +Plot D4, D5, D6, D7, and D8-plus contributions with a horizontal `10^-6` total target annotation. + +- [ ] **Step 4: Generate D4 norm reduction** + +Plot direct Pauli-l1 versus grouped anticommuting bound and annotate the exact reduction factor. + +- [ ] **Step 5: Generate the fivefold bottleneck figure** + +Use the committed audit values at `r=78`; label the entire panel "conditional feasibility audit, not a global certificate." + +- [ ] **Step 6: Write results and ablations** + +Cover the main result, adjacent boundary, D4 grouping, local lemma, recursive Suzuki audit, degree-three rank, three-site cluster result, small-system cross-check, and D5 side study. + +- [ ] **Step 7: Write limitations** + +Explain the single-model limitation, constant-factor rather than asymptotic improvement, incomplete comparison with optimized formulas, D4/D6/tail gates for fivefold, and order-lift requirement for tenfold. + +- [ ] **Step 8: Regenerate and validate figures** + +Run: + +```bash +python docs/manuscript/scripts/generate_figures.py +python docs/manuscript/scripts/validate_claims.py +``` + +Expected: four figure PDFs and `headline_claims=PASS`. + +### Task 7: Write reproducibility, conclusion, and paper metadata + +**Files:** +- Modify: `docs/manuscript/sections/reproducibility.tex` +- Modify: `docs/manuscript/sections/conclusion.tex` +- Modify: `docs/manuscript/main.tex` + +**Interfaces:** +- Consumes: README reproduction commands, transcript metadata, manifest hashes. +- Produces: a reviewer-executable handoff and compliant manuscript metadata. + +- [ ] **Step 1: Add the claim-to-evidence matrix** + +Map baseline, `r=97`, `r=96`, D4 partition, D5 sidecar, small cross-check, report, and manifest to exact paths and commands. + +- [ ] **Step 2: Add reproduction commands** + +Include environment installation, tests, fast verification, deep verification, D5 verification, package check, and checksum validation. + +- [ ] **Step 3: Add availability and contribution statements** + +Include Data Availability, Code Availability, Author Contributions, Acknowledgments, and an AI-assistance disclosure stating the actual role of language models in drafting, code assistance, and literature organization while assigning scientific responsibility to the author. + +- [ ] **Step 4: Write the conclusion** + +Restate the proof-compiler contribution, scoped best-known certificate, and complementary relationship to MPF/LCU/observable-specific methods. + +### Task 8: Compile, audit, and freeze the manuscript PDF + +**Files:** +- Produce: `docs/manuscript/output/pdf/issue128-proof-carrying-trotter-paper.pdf` +- Produce temporarily: `tmp/pdfs/issue128-paper/page-*.png` + +**Interfaces:** +- Consumes: all manuscript sources and generated figures. +- Produces: the final validated PDF and a clean audit report. + +- [ ] **Step 1: Compile with latexmk** + +Run: + +```bash +cd docs/manuscript +latexmk -pdf -interaction=nonstopmode -halt-on-error main.tex +mkdir -p output/pdf +cp main.pdf output/pdf/issue128-proof-carrying-trotter-paper.pdf +``` + +- [ ] **Step 2: Run structural checks** + +Run: + +```bash +pdfinfo output/pdf/issue128-proof-carrying-trotter-paper.pdf +pdftotext output/pdf/issue128-proof-carrying-trotter-paper.pdf - | wc -w +rg -n "undefined|Overfull|Underfull|multiply defined" main.log +``` + +Expected: at least 15 pages, no unresolved citation/reference diagnostics, and no material overfull boxes. + +- [ ] **Step 3: Scan for claim and drafting hazards** + +Run searches for placeholder markers, unqualified fivefold claims, accidental "published 393-step result" wording, and unqualified global-SOTA language. Correct every occurrence in context. + +- [ ] **Step 4: Render every page** + +Run: + +```bash +mkdir -p tmp/pdfs/issue128-paper +pdftoppm -png -r 120 output/pdf/issue128-proof-carrying-trotter-paper.pdf tmp/pdfs/issue128-paper/page +``` + +Inspect every rendered page for clipping, overlap, broken equations, illegible plots, excessive whitespace, bad table breaks, and orphaned headings. + +- [ ] **Step 5: Re-run evidence checks** + +Run: + +```bash +python scripts/validate_claims.py +cd ../.. +PYTHONPATH=src python scripts/verify.py certificates/issue128-certificate.json +PYTHONPATH=src python scripts/package_delivery.py --check +shasum -a 256 -c artifacts/SHA256SUMS +``` + +Expected: manuscript claims pass, certificate valid, delivery check passes, and all ten frozen hashes are unchanged. + +- [ ] **Step 6: Report the final artifact** + +Record page count, word count, PDF SHA-256, compiler command, and any remaining non-fatal typography warnings in the handoff. State explicitly that the manuscript is a local follow-up and has not been pushed to the frozen judged PR. + +## Self-review + +- Spec coverage: all requested detailed explanations, implementation guidance, long-form PDF, complete algorithm chain, and verification steps are assigned to Tasks 1--8. +- Placeholder scan: the plan contains no unresolved implementation placeholders. +- Interface consistency: manuscript paths, script names, figure names, and final PDF name are used consistently across tasks. +- Scope integrity: no task modifies frozen certificate artifacts or incorporates unrelated HPC work. diff --git a/tracks/qcs/solutions/WangTheoPhys/issue128/pyproject.toml b/tracks/qcs/solutions/WangTheoPhys/issue128/pyproject.toml new file mode 100644 index 000000000..af5ec0e76 --- /dev/null +++ b/tracks/qcs/solutions/WangTheoPhys/issue128/pyproject.toml @@ -0,0 +1,22 @@ +[build-system] +requires = ["setuptools>=70"] +build-backend = "setuptools.build_meta" + +[project] +name = "trottercert" +version = "0.1.0" +description = "Machine-certified product-formula analysis for 2D Heisenberg models" +requires-python = ">=3.11" +dependencies = ["numpy>=2.0", "scipy>=1.13", "sympy>=1.13"] + +[project.optional-dependencies] +test = ["pytest>=8"] + +[tool.setuptools] +package-dir = {"" = "src"} + +[tool.pytest.ini_options] +pythonpath = ["src"] +testpaths = ["tests"] +addopts = "-m 'not slow'" +markers = ["slow: long-running research reproduction test"] diff --git a/tracks/qcs/solutions/WangTheoPhys/issue128/scripts/__init__.py b/tracks/qcs/solutions/WangTheoPhys/issue128/scripts/__init__.py new file mode 100644 index 000000000..d78bd2fb2 --- /dev/null +++ b/tracks/qcs/solutions/WangTheoPhys/issue128/scripts/__init__.py @@ -0,0 +1 @@ +"""Importable command-line helpers for the Issue 128 submission.""" diff --git a/tracks/qcs/solutions/WangTheoPhys/issue128/scripts/build_d5_certificate.py b/tracks/qcs/solutions/WangTheoPhys/issue128/scripts/build_d5_certificate.py new file mode 100644 index 000000000..b95727327 --- /dev/null +++ b/tracks/qcs/solutions/WangTheoPhys/issue128/scripts/build_d5_certificate.py @@ -0,0 +1,131 @@ +#!/usr/bin/env python3 +from __future__ import annotations + +import argparse +from fractions import Fraction +from pathlib import Path + +from trottercert.cubic_field import fourth_order_suzuki_cubic_stages +from trottercert.cubic_local import exact_d5_density, exact_log_e5_density +from trottercert.intervals import cube_root_four_interval +from trottercert.support_groups import ( + build_d5_payload, + canonical_gzip_bytes, + certify_support_partition, + decode_d5_gzip, + discover_support_partition, + verify_d5_payload, +) + + +ROOT = Path(__file__).resolve().parents[1] +OUTPUT = ROOT / "certificates" / "issue128-d5-groups.json.gz" +EXPECTED_TERM_COUNT = 605_832 +EXPECTED_GROUP_COUNT = 123_106 +EXPECTED_SITE_DENSITY = Fraction( + 44_948_270_001_027_856_175_670_154_896_253, + 4_000_000_000_000_000_000_000_000_000_000, +) +SITE_DENSITY_GATE = Fraction(12) + + +def _require_regression_gates( + *, + term_count: int, + group_count: int, + site_density: Fraction, +) -> None: + if term_count != EXPECTED_TERM_COUNT: + raise ValueError( + f"D5 term-count regression: {term_count} != {EXPECTED_TERM_COUNT}" + ) + if group_count != EXPECTED_GROUP_COUNT: + raise ValueError( + f"D5 group-count regression: {group_count} != {EXPECTED_GROUP_COUNT}" + ) + if site_density > SITE_DENSITY_GATE: + raise ValueError( + f"D5 site-density gate failed: {site_density} > {SITE_DENSITY_GATE}" + ) + if site_density != EXPECTED_SITE_DENSITY: + raise ValueError( + "D5 site-density regression: " + f"{site_density} != {EXPECTED_SITE_DENSITY}" + ) + + +def _report(term_count: int, group_count: int, site_density: Fraction) -> None: + print(f"term_count={term_count}") + print(f"group_count={group_count}") + print(f"site_density={site_density}") + print(f"site_density_decimal={float(site_density):.16g}") + + +def verify_only() -> None: + raw = OUTPUT.read_bytes() + payload = decode_d5_gzip(raw) + certificate = verify_d5_payload(payload) + term_count = len(certificate.paulis) + group_count = len(certificate.groups) + site_density = certificate.bound / 4 + _require_regression_gates( + term_count=term_count, + group_count=group_count, + site_density=site_density, + ) + _report(term_count, group_count, site_density) + + +def build() -> None: + stages = fourth_order_suzuki_cubic_stages(4) + registry, e5 = exact_log_e5_density(stages) + d5 = exact_d5_density(registry, e5) + root = cube_root_four_interval(24) + root_midpoint = root.midpoint() + coefficients = { + pauli: coefficient.enclose(root) + for pauli, coefficient in d5.items() + } + discovery_scores = { + pauli: coefficient.a0 + + coefficient.a1 * root_midpoint + + coefficient.a2 * root_midpoint**2 + for pauli, coefficient in d5.items() + } + groups = discover_support_partition( + coefficients, + max_group_size=10, + discovery_scores=discovery_scores, + ) + certificate = certify_support_partition(coefficients, groups) + term_count = len(certificate.paulis) + group_count = len(certificate.groups) + site_density = certificate.bound / 4 + _report(term_count, group_count, site_density) + _require_regression_gates( + term_count=term_count, + group_count=group_count, + site_density=site_density, + ) + + payload = build_d5_payload(coefficients, groups) + raw = canonical_gzip_bytes(payload) + verified = verify_d5_payload(decode_d5_gzip(raw)) + if verified.bound != certificate.bound: + raise ValueError("D5 round-trip certificate bound mismatch") + OUTPUT.write_bytes(raw) + print(f"output={OUTPUT}") + + +def main() -> None: + parser = argparse.ArgumentParser() + parser.add_argument("--verify-only", action="store_true") + arguments = parser.parse_args() + if arguments.verify_only: + verify_only() + else: + build() + + +if __name__ == "__main__": + main() diff --git a/tracks/qcs/solutions/WangTheoPhys/issue128/scripts/build_d8_shard.py b/tracks/qcs/solutions/WangTheoPhys/issue128/scripts/build_d8_shard.py new file mode 100755 index 000000000..a60d9efba --- /dev/null +++ b/tracks/qcs/solutions/WangTheoPhys/issue128/scripts/build_d8_shard.py @@ -0,0 +1,164 @@ +#!/usr/bin/env python3 +from __future__ import annotations + +import argparse +from datetime import datetime, timezone +import json +import os +from pathlib import Path +import resource +import subprocess +import sys +import time +import traceback + +from trottercert.cubic_field import fourth_order_suzuki_cubic_stages +from trottercert.cubic_local import exact_right_generator_stage_contribution +from trottercert.hpc_artifacts import ( + coordinate_encode_series, + cubic_to_json, + sha256_file, + write_manifest_atomic, + write_shard_gzip, +) + + +FORMULA_ID = "five_copy_suzuki_fourth_order_exact_cubic" + + +def _utc_now() -> str: + return datetime.now(timezone.utc).isoformat() + + +def _source_state(root: Path) -> tuple[str, bool]: + shipped_commit = os.environ.get("ISSUE128_SOURCE_COMMIT") + if shipped_commit: + dirty_text = os.environ.get("ISSUE128_SOURCE_DIRTY", "0").lower() + return shipped_commit, dirty_text in {"1", "true", "yes"} + commit = subprocess.run( + ["git", "rev-parse", "HEAD"], + cwd=root, + check=True, + capture_output=True, + text=True, + ).stdout.strip() + dirty = bool( + subprocess.run( + ["git", "status", "--porcelain"], + cwd=root, + check=True, + capture_output=True, + text=True, + ).stdout.strip() + ) + return commit, dirty + + +def _max_rss_bytes() -> int: + value = resource.getrusage(resource.RUSAGE_SELF).ru_maxrss + # macOS reports bytes; Linux reports KiB. + return int(value if sys.platform == "darwin" else value * 1024) + + +def build_shard( + *, + stage_index: int, + order: int, + output: Path, + manifest: Path, + repository_root: Path, +) -> dict[str, object]: + stages = fourth_order_suzuki_cubic_stages() + if stage_index < 0 or stage_index >= len(stages): + raise ValueError(f"stage index must be in [0, {len(stages) - 1}]") + if order < 0: + raise ValueError("order must be nonnegative") + commit, dirty = _source_state(repository_root) + started_at = _utc_now() + started = time.perf_counter() + running = { + "schema_version": 1, + "kind": "issue128_d8_stage_manifest", + "status": "running", + "stage_index": stage_index, + "stage_count": len(stages), + "order": order, + "git_commit": commit, + "git_dirty": dirty, + "started_at": started_at, + } + write_manifest_atomic(manifest, running) + print( + f"stage={stage_index}/{len(stages)-1} order={order} status=running", + flush=True, + ) + try: + registry, series = exact_right_generator_stage_contribution( + stages, stage_index, order + ) + stage = stages[stage_index] + payload = { + "schema_version": 1, + "kind": "issue128_exact_right_generator_stage", + "formula_id": FORMULA_ID, + "stage_index": stage_index, + "stage_count": len(stages), + "fragment_index": stage.fragment_index, + "stage_coefficient": cubic_to_json(stage.coefficient), + "order": order, + "series": coordinate_encode_series(registry, series), + } + write_shard_gzip(output, payload) + elapsed = time.perf_counter() - started + complete = { + **running, + "status": "complete", + "completed_at": _utc_now(), + "wall_seconds": elapsed, + "peak_rss_bytes": _max_rss_bytes(), + "degree_term_counts": [len(degree) for degree in series], + "output": output.name, + "output_sha256": sha256_file(output), + } + write_manifest_atomic(manifest, complete) + print( + f"stage={stage_index} status=complete wall={elapsed:.3f}s " + f"terms={complete['degree_term_counts']} sha256={complete['output_sha256']}", + flush=True, + ) + return complete + except BaseException as error: + failed = { + **running, + "status": "failed", + "completed_at": _utc_now(), + "wall_seconds": time.perf_counter() - started, + "peak_rss_bytes": _max_rss_bytes(), + "error_type": type(error).__name__, + "error": str(error), + "traceback": traceback.format_exc(), + } + write_manifest_atomic(manifest, failed) + raise + + +def main() -> None: + parser = argparse.ArgumentParser() + parser.add_argument("--stage-index", type=int, required=True) + parser.add_argument("--order", type=int, default=8) + parser.add_argument("--output", type=Path, required=True) + parser.add_argument("--manifest", type=Path, required=True) + arguments = parser.parse_args() + issue_root = Path(__file__).resolve().parents[1] + repository_root = issue_root.parents[4] + build_shard( + stage_index=arguments.stage_index, + order=arguments.order, + output=arguments.output, + manifest=arguments.manifest, + repository_root=repository_root, + ) + + +if __name__ == "__main__": + main() diff --git a/tracks/qcs/solutions/WangTheoPhys/issue128/scripts/build_v3_certificate.py b/tracks/qcs/solutions/WangTheoPhys/issue128/scripts/build_v3_certificate.py new file mode 100644 index 000000000..bb917fed0 --- /dev/null +++ b/tracks/qcs/solutions/WangTheoPhys/issue128/scripts/build_v3_certificate.py @@ -0,0 +1,283 @@ +#!/usr/bin/env python3 +from __future__ import annotations + +import hashlib +import json +from fractions import Fraction +from math import lcm +from pathlib import Path + +from trottercert.anticommuting import ( + AnticommutingPartitionCertificate, + certify_anticommuting_partition, + discover_anticommuting_partition, +) +from trottercert.pareto import minimum_published_suzuki_point +from trottercert.refined_error import ( + build_refined_fourth_order_constants, + certified_d4_cell_coefficients, + evaluate_refined_fourth_order_bound, +) +from trottercert.resources import required_steps +from trottercert.rigorous_fourth import ( + fourth_order_published_triangle_certificate, +) + + +ROOT = Path(__file__).resolve().parents[1] +OUTPUT = ROOT / "certificates" / "issue128-certificate.json" +D4_OUTPUT = ROOT / "certificates" / "issue128-d4-groups.json" + + +def pair(value: Fraction) -> list[int]: + return [value.numerator, value.denominator] + + +def canonical_json_bytes(payload: dict[str, object]) -> bytes: + return ( + json.dumps(payload, sort_keys=True, separators=(",", ":")) + + "\n" + ).encode() + + +def d4_sidecar_payload( + certificate: AnticommutingPartitionCertificate, +) -> dict[str, object]: + coefficient_denominator = 1 + for interval in certificate.coefficients: + coefficient_denominator = lcm( + coefficient_denominator, + interval.lower.denominator, + interval.upper.denominator, + ) + sqrt_denominator = 1 + for group in certificate.groups: + sqrt_denominator = lcm( + sqrt_denominator, + group.bound.denominator, + ) + return { + "schema_version": 1, + "coefficient_denominator": coefficient_denominator, + "terms": [ + [ + pauli[0], + pauli[1], + interval.lower.numerator + * ( + coefficient_denominator + // interval.lower.denominator + ), + interval.upper.numerator + * ( + coefficient_denominator + // interval.upper.denominator + ), + ] + for pauli, interval in zip( + certificate.paulis, + certificate.coefficients, + ) + ], + "sqrt_denominator": sqrt_denominator, + "groups": [ + [ + list(group.term_indices), + group.bound.numerator + * (sqrt_denominator // group.bound.denominator), + ] + for group in certificate.groups + ], + "cell_bound": pair(certificate.bound), + } + + +def main() -> None: + n_sites = 144 + tolerance = Fraction(1, 10**6) + + published = fourth_order_published_triangle_certificate( + center=20, + decimal_digits=18, + ) + published_steps = required_steps( + published.site_density_upper * n_sites, + tolerance, + 4, + ) + published_groups = 30 * published_steps + 1 + + constants = build_refined_fourth_order_constants( + decimal_digits=12, + quantization_digits=18, + ) + d4_coefficients = certified_d4_cell_coefficients( + constants.stages, + quantization_digits=18, + ) + d4_groups = discover_anticommuting_partition( + d4_coefficients, + max_group_size=10, + ) + d4_partition = certify_anticommuting_partition( + d4_coefficients, + d4_groups, + ) + d4_payload = d4_sidecar_payload(d4_partition) + d4_bytes = canonical_json_bytes(d4_payload) + D4_OUTPUT.write_bytes(d4_bytes) + d4_site_bound = d4_partition.bound / 4 + + low = 1 + high = published_steps + while low < high: + middle = (low + high) // 2 + if ( + evaluate_refined_fourth_order_bound( + constants, + n_sites, + middle, + d4_site_override=d4_site_bound, + ).global_error_bound + <= tolerance + ): + high = middle + else: + low = middle + 1 + candidate_steps = low + candidate = evaluate_refined_fourth_order_bound( + constants, + n_sites, + candidate_steps, + d4_site_override=d4_site_bound, + ) + previous = evaluate_refined_fourth_order_bound( + constants, + n_sites, + candidate_steps - 1, + d4_site_override=d4_site_bound, + ) + candidate_groups = 30 * candidate_steps + 1 + + recursive_points = [] + for order in (2, 4, 6, 8): + point = minimum_published_suzuki_point( + order, + n_sites=n_sites, + tolerance=tolerance, + decimal_digits=12, + ) + recursive_points.append( + { + "order": point.order, + "steps": point.steps, + "stages_per_step": point.stages_per_step, + "group_exponentials": point.group_exponentials, + "global_error_upper_decimal_summary": format( + float(point.global_error_bound), + ".17g", + ), + } + ) + + baseline_bonds = published_groups * n_sites // 2 + candidate_bonds = candidate_groups * n_sites // 2 + certificate = { + "schema_version": 3, + "benchmark": { + "model": "periodic_square_spin_half_isotropic_heisenberg", + "normalization": "(XX+YY+ZZ)/4", + "length": 12, + "time": [1, 1], + "tolerance": [1, 10**6], + "primary_certified_metric": "compiled_cnot_upper", + }, + "published_baseline": { + "source_theorem": "Childs_Su_Tran_Wiebe_Zhu_high_order_commutator_bound", + "formula": "five_copy_suzuki_fourth_order", + "formula_order": 4, + "stage_count": 31, + "theorem_center": 20, + "coefficient_interval_decimal_digits": 18, + "norm_method": "expand_partial_sums_then_local_Pauli_l1", + "site_density_upper": pair(published.site_density_upper), + "theorem_terms": published.theorem_terms, + "expanded_commutator_keys": published.expanded_commutator_keys, + "steps": published_steps, + "group_exponentials": published_groups, + }, + "candidate": { + "formula": "five_copy_suzuki_fourth_order", + "proof_method": ( + "local_log_E5_grouped_D4_plus_E7_majorant" + "_plus_exact_generator_tail" + ), + "coefficient_interval_decimal_digits": 12, + "e5_quantization_digits": 18, + "e5_site_l1_upper": pair(constants.e5_site_l1), + "e7_site_majorant": pair(constants.e7_site_majorant), + "d4_certificate": { + "path": D4_OUTPUT.name, + "sha256": hashlib.sha256(d4_bytes).hexdigest(), + "max_group_size": 10, + "term_count": len(d4_partition.paulis), + "group_count": len(d4_partition.groups), + "cell_norm_upper": pair(d4_partition.bound), + }, + "steps": candidate_steps, + "group_exponentials": candidate_groups, + "contributions": { + "degree4": pair(candidate.degree_four_contribution), + "degree5": pair(candidate.degree_five_contribution), + "degree6": pair(candidate.degree_six_contribution), + "degree7": pair(candidate.degree_seven_contribution), + "tail": pair(candidate.tail_contribution), + }, + "global_error_upper": pair(candidate.global_error_bound), + "previous_step_error_upper": pair(previous.global_error_bound), + }, + "cost_model": { + "bonds_per_matching": n_sites // 2, + "cnots_per_bond_propagator": 3, + "inter_step_merge_rule": "30*r+1", + }, + "claimed_resources": { + "published_steps": published_steps, + "candidate_steps": candidate_steps, + "published_group_exponentials": published_groups, + "candidate_group_exponentials": candidate_groups, + "published_bond_propagators": baseline_bonds, + "candidate_bond_propagators": candidate_bonds, + "published_cnot_upper": 3 * baseline_bonds, + "candidate_cnot_upper": 3 * candidate_bonds, + }, + "published_recursive_suzuki_audit": recursive_points, + "claims": { + "global_twofold_target_met": published_groups >= 2 * candidate_groups, + "global_fourfold_target_met": ( + published_groups >= 4 * candidate_groups + ), + "exact_improvement_ratio": pair( + Fraction(published_groups, candidate_groups) + ), + }, + } + OUTPUT.write_text(json.dumps(certificate, indent=2) + "\n") + print(D4_OUTPUT) + print(OUTPUT) + print( + json.dumps( + { + "published_groups": published_groups, + "candidate_groups": candidate_groups, + "improvement": published_groups / candidate_groups, + "candidate_error": float(candidate.global_error_bound), + "previous_error": float(previous.global_error_bound), + }, + indent=2, + ) + ) + + +if __name__ == "__main__": + main() diff --git a/tracks/qcs/solutions/WangTheoPhys/issue128/scripts/crosscheck_small.py b/tracks/qcs/solutions/WangTheoPhys/issue128/scripts/crosscheck_small.py new file mode 100644 index 000000000..ed962f2db --- /dev/null +++ b/tracks/qcs/solutions/WangTheoPhys/issue128/scripts/crosscheck_small.py @@ -0,0 +1,25 @@ +#!/usr/bin/env python3 +from __future__ import annotations + +import argparse +import json +from fractions import Fraction + +from trottercert.crosscheck import small_exact_crosscheck + + +def main() -> None: + parser = argparse.ArgumentParser() + parser.add_argument("--length", type=int, default=2) + parser.add_argument("--tolerance", type=int, default=10**6) + args = parser.parse_args() + print( + json.dumps( + small_exact_crosscheck(args.length, Fraction(1, args.tolerance)), + indent=2, + ) + ) + + +if __name__ == "__main__": + main() diff --git a/tracks/qcs/solutions/WangTheoPhys/issue128/scripts/package_delivery.py b/tracks/qcs/solutions/WangTheoPhys/issue128/scripts/package_delivery.py new file mode 100644 index 000000000..9aeabb24d --- /dev/null +++ b/tracks/qcs/solutions/WangTheoPhys/issue128/scripts/package_delivery.py @@ -0,0 +1,426 @@ +#!/usr/bin/env python3 +"""Build and validate the reviewer-facing Issue 128 delivery package.""" + +from __future__ import annotations + +import argparse +from decimal import Decimal, ROUND_CEILING, localcontext +import gzip +import hashlib +import json +import os +from pathlib import Path +import platform +import subprocess +import sys +from typing import Mapping + + +ROOT = Path(__file__).resolve().parents[1] +CERTIFICATE = Path("certificates/issue128-certificate.json") +D4_SIDECAR = Path("certificates/issue128-d4-groups.json") +D5_SIDECAR = Path("certificates/issue128-d5-groups.json.gz") +SMALL_CROSSCHECK = Path("certificates/issue128-small-crosscheck.json") + +EXPECTED: dict[str, object] = { + "published_steps": 393, + "candidate_steps": 97, + "published_groups": 11_791, + "candidate_groups": 2_911, + "ratio": [11_791, 2_911], + "d4_terms": 75_324, + "d4_groups": 7_576, + "d5_terms": 605_832, + "d5_groups": 123_106, + "certificate_sha256": "0a09623ce3b292a3637065c870fb3153bbdcddce30aef968565c4db3ddfc7201", + "d4_sha256": "a397414bb0229fb1ebdb38798aa781fb89dbb9d5cdbed94c7cd2e9120da62718", + "d5_sha256": "c5e8968a93b4497b41fe42c0e364324388272e183e1fcd20a536bc988f5361dd", + "pr_head": "c550a6b0915ccf7a1db410ee765ff65d99f96b6f", +} + +MANIFEST_PATHS = ( + Path("artifacts/issue128-summary.json"), + Path("artifacts/issue128-summary.txt"), + Path("artifacts/verification-transcript.txt"), + CERTIFICATE, + D4_SIDECAR, + D5_SIDECAR, + SMALL_CROSSCHECK, + Path("docs/report/issue128-technical-report.tex"), + Path("docs/report/references.bib"), + Path("docs/report/output/issue128-technical-report.pdf"), +) + + +def require_equal(label: str, actual: object, expected: object) -> None: + if actual != expected: + raise ValueError(f"{label} drift: {actual!r} != {expected!r}") + + +def canonical_json(payload: object) -> str: + return json.dumps(payload, indent=2, sort_keys=True) + "\n" + + +def sha256_file(path: Path) -> str: + digest = hashlib.sha256() + with path.open("rb") as handle: + for chunk in iter(lambda: handle.read(1 << 20), b""): + digest.update(chunk) + return digest.hexdigest() + + +def _object(path: Path) -> dict[str, object]: + payload = json.loads(path.read_text()) + if not isinstance(payload, dict): + raise TypeError(f"expected an object in {path}") + return payload + + +def _gzip_object(path: Path) -> dict[str, object]: + with gzip.open(path, "rt") as handle: + payload = json.load(handle) + if not isinstance(payload, dict): + raise TypeError(f"expected an object in {path}") + return payload + + +def _decimal(pair: list[int], digits: int = 16) -> str: + with localcontext() as context: + context.prec = 80 + value = Decimal(pair[0]) / Decimal(pair[1]) + return format(value, f".{digits}g").lower() + + +def _outward_upper(pair: list[int], digits: int = 16) -> str: + """Format a positive rational upward, never below the certified bound.""" + + with localcontext() as context: + context.prec = 100 + value = Decimal(pair[0]) / Decimal(pair[1]) + exponent = value.adjusted() + quantum = Decimal(1).scaleb(exponent - digits + 1) + rounded = value.quantize(quantum, rounding=ROUND_CEILING) + return format(rounded, f".{digits - 1}e").lower() + + +def _git_provenance(root: Path) -> tuple[str, str]: + repository = subprocess.run( + ["git", "rev-parse", "--show-toplevel"], + cwd=root, + check=True, + capture_output=True, + text=True, + ).stdout.strip() + relative = (root / CERTIFICATE).resolve().relative_to(Path(repository).resolve()) + output = subprocess.run( + ["git", "log", "-1", "--format=%H%n%cI", "--", relative.as_posix()], + cwd=repository, + check=True, + capture_output=True, + text=True, + ).stdout.strip().splitlines() + if len(output) != 2: + raise ValueError("could not determine certificate provenance") + return output[0], output[1] + + +def build_summary(root: Path = ROOT) -> dict[str, object]: + root = root.resolve() + certificate = _object(root / CERTIFICATE) + crosscheck = _object(root / SMALL_CROSSCHECK) + d5 = _gzip_object(root / D5_SIDECAR) + benchmark = certificate["benchmark"] + published = certificate["published_baseline"] + candidate = certificate["candidate"] + resources = certificate["claimed_resources"] + claims = certificate["claims"] + if not all(isinstance(item, dict) for item in (benchmark, published, candidate, resources, claims)): + raise TypeError("main certificate schema is malformed") + + checks = ( + ("published steps", resources["published_steps"], EXPECTED["published_steps"]), + ("candidate steps", resources["candidate_steps"], EXPECTED["candidate_steps"]), + ("published groups", resources["published_group_exponentials"], EXPECTED["published_groups"]), + ("candidate groups", resources["candidate_group_exponentials"], EXPECTED["candidate_groups"]), + ("exact ratio", claims["exact_improvement_ratio"], EXPECTED["ratio"]), + ) + for label, actual, expected in checks: + require_equal(label, actual, expected) + + d4 = candidate["d4_certificate"] + if not isinstance(d4, dict): + raise TypeError("D4 metadata is malformed") + require_equal("D4 terms", d4["term_count"], EXPECTED["d4_terms"]) + require_equal("D4 groups", d4["group_count"], EXPECTED["d4_groups"]) + require_equal("D5 terms", d5["term_count"], EXPECTED["d5_terms"]) + require_equal("D5 groups", d5["group_count"], EXPECTED["d5_groups"]) + + hashes = { + "main_certificate": sha256_file(root / CERTIFICATE), + "d4_sidecar": sha256_file(root / D4_SIDECAR), + "d5_sidecar": sha256_file(root / D5_SIDECAR), + } + require_equal("main certificate digest", hashes["main_certificate"], EXPECTED["certificate_sha256"]) + require_equal("D4 sidecar digest", hashes["d4_sidecar"], EXPECTED["d4_sha256"]) + require_equal("D5 sidecar digest", hashes["d5_sidecar"], EXPECTED["d5_sha256"]) + + tolerance = benchmark["tolerance"] + error = candidate["global_error_upper"] + previous_error = candidate["previous_step_error_upper"] + accepted = error[0] * tolerance[1] <= tolerance[0] * error[1] + rejected = previous_error[0] * tolerance[1] > tolerance[0] * previous_error[1] + require_equal("97-step acceptance", accepted, True) + require_equal("96-step rejection", rejected, True) + + source_commit, source_timestamp = _git_provenance(root) + exact_ratio = claims["exact_improvement_ratio"] + conditional_ratio = [resources["published_group_exponentials"], 30 * 78 + 1] + removed = [ + resources["published_group_exponentials"] - resources["candidate_group_exponentials"], + resources["published_group_exponentials"], + ] + contributions = candidate["contributions"] + if not isinstance(contributions, dict): + raise TypeError("error contribution ledger is malformed") + + return { + "schema_version": 1, + "sources": { + "anchored_pr_head": EXPECTED["pr_head"], + "source_commit_for_main_certificate": source_commit, + "source_commit_timestamp": source_timestamp, + "main_certificate": {"path": CERTIFICATE.as_posix(), "sha256": hashes["main_certificate"]}, + "d4_sidecar": {"path": D4_SIDECAR.as_posix(), "sha256": hashes["d4_sidecar"]}, + "d5_sidecar": {"path": D5_SIDECAR.as_posix(), "sha256": hashes["d5_sidecar"]}, + }, + "benchmark": { + "model": benchmark["model"], + "normalization": benchmark["normalization"], + "length": benchmark["length"], + "sites": benchmark["length"] ** 2, + "time": benchmark["time"], + "tolerance": tolerance, + "formula": candidate["formula"], + "formula_order": published["formula_order"], + "stage_count": published["stage_count"], + }, + "published_baseline": { + "source_theorem": published["source_theorem"], + "theorem_center": published["theorem_center"], + "site_density_upper": published["site_density_upper"], + "site_density_upper_decimal": _outward_upper(published["site_density_upper"]), + "steps": resources["published_steps"], + "group_exponentials": resources["published_group_exponentials"], + "bond_propagators": resources["published_bond_propagators"], + "cnot_upper": resources["published_cnot_upper"], + }, + "certified_result": { + "status": "certified", + "steps": resources["candidate_steps"], + "group_exponentials": resources["candidate_group_exponentials"], + "bond_propagators": resources["candidate_bond_propagators"], + "cnot_upper": resources["candidate_cnot_upper"], + "error_upper": error, + "error_upper_decimal_outward": _outward_upper(error), + "previous_step": 96, + "previous_step_error_upper": previous_error, + "previous_step_error_upper_decimal_outward": _outward_upper(previous_error), + "accepted_at_97": accepted, + "rejected_at_96": rejected, + }, + "improvement": { + "exact_ratio": exact_ratio, + "decimal": _decimal(exact_ratio), + "resource_fraction_removed": removed, + "resource_percent_removed": _decimal([100 * removed[0], removed[1]]) + "%", + "multiple_of_required_twofold_target": _decimal([exact_ratio[0], 2 * exact_ratio[1]]), + }, + "error_ledger": { + name: {"exact": pair, "decimal_outward": _outward_upper(pair)} + for name, pair in contributions.items() + }, + "method": { + "proof_method": candidate["proof_method"], + "innovations": [ + "full free-associative-word logarithm before norm inequalities", + "Dynkin-Specht-Wever projection to degree-5 and degree-7 Lie elements", + "translation-canonical Pauli combination before taking norms", + "exact pairwise-anticommuting grouping for the leading D4 defect", + "exact single-bond Heisenberg commutator growth constant 1", + "finite-step right-generator ledger with a rational geometric tail", + "independent exact-rational and symplectic verification", + ], + }, + "verification": { + "d4_term_count": d4["term_count"], + "d4_group_count": d4["group_count"], + "d4_max_group_size": d4["max_group_size"], + "d4_cell_norm_upper": d4["cell_norm_upper"], + "d4_cell_norm_upper_decimal": _outward_upper(d4["cell_norm_upper"]), + "small_crosscheck": crosscheck, + }, + "fivefold_followup": { + "status": "not_certified", + "conditional_step_count": 78, + "conditional_group_exponentials": conditional_ratio[1], + "conditional_exact_ratio": conditional_ratio, + "conditional_ratio_decimal": _decimal(conditional_ratio), + "d5_term_count": d5["term_count"], + "d5_group_count": d5["group_count"], + "d5_site_density_upper": d5["site_bound"], + "d5_site_density_upper_decimal": _outward_upper(d5["site_bound"]), + "unresolved_proof_gates": [ + "translation-coupled D4 norm below the r=78 budget", + "explicit D8 and delayed high-degree tail certificate", + "complete exact global error ledger at r=78", + ], + "claim_boundary": "No 78-step global error certificate is claimed or supplied.", + }, + } + + +def render_summary_text(summary: Mapping[str, object]) -> str: + b = summary["benchmark"] + p = summary["published_baseline"] + r = summary["certified_result"] + i = summary["improvement"] + v = summary["verification"] + f = summary["fivefold_followup"] + s = summary["sources"] + return "\n".join(( + "ISSUE 128 FINAL RESULT SUMMARY", + "================================", + "", + f"BENCHMARK: {b['model']}", + f"LATTICE: L={b['length']}, N={b['sites']}", + f"NORMALIZATION: {b['normalization']}", + "TIME: T=1; OPERATOR-NORM TOLERANCE: 1e-6", + "", + f"PUBLISHED BASELINE: {p['steps']} steps, {p['group_exponentials']} groups, {p['bond_propagators']} bond propagators, {p['cnot_upper']} CNOT upper", + f"CERTIFIED RESULT: {i['decimal']}x", + f" {r['steps']} steps, {r['group_exponentials']} groups, {r['bond_propagators']} bond propagators, {r['cnot_upper']} CNOT upper", + f" exact ratio: {i['exact_ratio'][0]}/{i['exact_ratio'][1]}", + f" resource reduction: {i['resource_percent_removed']}", + f" 97-step error upper (outward decimal): {r['error_upper_decimal_outward']}", + f" 96-step error upper (outward decimal): {r['previous_step_error_upper_decimal_outward']}", + " integer boundary: 97 accepted; 96 rejected", + "", + f"EXACT D4: {v['d4_term_count']} terms, {v['d4_group_count']} groups, maximum size {v['d4_max_group_size']}, cell bound {v['d4_cell_norm_upper_decimal']}", + "", + "FIVEFOLD STATUS: NOT CERTIFIED", + f" conditional arithmetic only: r={f['conditional_step_count']}, groups={f['conditional_group_exponentials']}, ratio={f['conditional_ratio_decimal']}x", + f" verified D5 evidence: {f['d5_term_count']} terms, {f['d5_group_count']} groups, site bound {f['d5_site_density_upper_decimal']}", + f" {f['claim_boundary']}", + "", + "SOURCE BINDINGS", + f" main certificate SHA-256: {s['main_certificate']['sha256']}", + f" D4 sidecar SHA-256: {s['d4_sidecar']['sha256']}", + f" D5 sidecar SHA-256: {s['d5_sidecar']['sha256']}", + "", + )) + + +def _run(root: Path, arguments: list[str]) -> subprocess.CompletedProcess[str]: + environment = dict(os.environ) + environment["PYTHONPATH"] = "src" + return subprocess.run([sys.executable, *arguments], cwd=root, env=environment, capture_output=True, text=True) + + +def capture_verification(root: Path = ROOT) -> str: + root = root.resolve() + commit, timestamp = _git_provenance(root) + lines = [ + "ISSUE 128 VERIFICATION TRANSCRIPT", + "=================================", + f"source_commit_for_main_certificate: {commit}", + f"source_commit_timestamp: {timestamp}", + f"python: {platform.python_version()}", + f"platform: {platform.platform()}", + "", + ] + for arguments in ( + ["scripts/verify.py", CERTIFICATE.as_posix()], + ["scripts/build_d5_certificate.py", "--verify-only"], + ): + completed = _run(root, arguments) + display = "PYTHONPATH=src python " + " ".join(arguments) + lines.extend(( + f"$ {display}", + f"exit_code: {completed.returncode}", + "--- stdout ---", + completed.stdout.rstrip(), + "--- stderr ---", + completed.stderr.rstrip() or "(empty)", + "", + )) + if completed.returncode: + raise RuntimeError(f"verification command failed: {display}") + lines.extend(("OVERALL STATUS: PASS", "")) + return "\n".join(lines) + + +def write_delivery_files(root: Path = ROOT, output_dir: Path | None = None, *, capture: bool = True) -> set[Path]: + root = root.resolve() + destination = output_dir or root / "artifacts" + destination.mkdir(parents=True, exist_ok=True) + summary = build_summary(root) + json_path = destination / "issue128-summary.json" + text_path = destination / "issue128-summary.txt" + json_path.write_text(canonical_json(summary)) + text_path.write_text(render_summary_text(summary)) + written = {json_path, text_path} + if capture: + transcript = destination / "verification-transcript.txt" + transcript.write_text(capture_verification(root)) + written.add(transcript) + return written + + +def write_sha_manifest(root: Path = ROOT) -> Path: + root = root.resolve() + missing = [path.as_posix() for path in MANIFEST_PATHS if not (root / path).is_file()] + if missing: + raise FileNotFoundError("manifest inputs missing: " + ", ".join(missing)) + manifest = root / "artifacts/SHA256SUMS" + manifest.write_text("\n".join(f"{sha256_file(root / path)} {path.as_posix()}" for path in MANIFEST_PATHS) + "\n") + return manifest + + +def verify_sha_manifest(root: Path = ROOT) -> None: + root = root.resolve() + for line in (root / "artifacts/SHA256SUMS").read_text().splitlines(): + digest, relative = line.split(" ", 1) + require_equal(f"manifest digest for {relative}", sha256_file(root / relative), digest) + + +def check_delivery(root: Path = ROOT) -> None: + root = root.resolve() + summary = build_summary(root) + require_equal("summary JSON", (root / "artifacts/issue128-summary.json").read_text(), canonical_json(summary)) + require_equal("summary text", (root / "artifacts/issue128-summary.txt").read_text(), render_summary_text(summary)) + if "OVERALL STATUS: PASS" not in capture_verification(root): + raise ValueError("current verifier run did not pass") + if "OVERALL STATUS: PASS" not in (root / "artifacts/verification-transcript.txt").read_text(): + raise ValueError("committed verifier transcript did not pass") + verify_sha_manifest(root) + + +def main() -> None: + parser = argparse.ArgumentParser() + parser.add_argument("--output-dir", type=Path) + parser.add_argument("--manifest", action="store_true") + parser.add_argument("--check", action="store_true") + arguments = parser.parse_args() + if arguments.check: + check_delivery(ROOT) + print("delivery_check=PASS") + return + written = write_delivery_files(ROOT, arguments.output_dir) + if arguments.manifest: + written.add(write_sha_manifest(ROOT)) + for path in sorted(written): + print(f"wrote={path}") + + +if __name__ == "__main__": + main() diff --git a/tracks/qcs/solutions/WangTheoPhys/issue128/scripts/reduce_d8_shards.py b/tracks/qcs/solutions/WangTheoPhys/issue128/scripts/reduce_d8_shards.py new file mode 100755 index 000000000..21dfbf35b --- /dev/null +++ b/tracks/qcs/solutions/WangTheoPhys/issue128/scripts/reduce_d8_shards.py @@ -0,0 +1,137 @@ +#!/usr/bin/env python3 +from __future__ import annotations + +import argparse +from decimal import Decimal, localcontext +from fractions import Fraction +import json +from pathlib import Path + +from trottercert.hpc_artifacts import ( + coordinate_terms_to_json, + merge_coordinate_series, + read_shard_gzip, + sha256_file, + write_manifest_atomic, + write_shard_gzip, +) +from trottercert.intervals import cube_root_four_interval + + +def _pair(value: Fraction) -> list[int]: + return [value.numerator, value.denominator] + + +def _decimal(value: Fraction, digits: int = 18) -> str: + with localcontext() as context: + context.prec = digits + return str(Decimal(value.numerator) / Decimal(value.denominator)) + + +def reduce_shards( + input_root: Path, + *, + expected_stages: int, + order: int, + output: Path, + summary_path: Path, +) -> dict[str, object]: + payloads: dict[int, dict[str, object]] = {} + commits: set[str] = set() + for stage in range(expected_stages): + cell = input_root / f"stage-{stage:02d}" + payload_path = cell / "shard.json.gz" + manifest_path = cell / "manifest.json" + if not payload_path.is_file() or not manifest_path.is_file(): + raise ValueError(f"missing shard or manifest for stage {stage}") + manifest = json.loads(manifest_path.read_text()) + if manifest.get("status") != "complete": + raise ValueError(f"stage {stage} manifest is not complete") + if manifest.get("output_sha256") != sha256_file(payload_path): + raise ValueError(f"stage {stage} payload digest mismatch") + payload = read_shard_gzip(payload_path) + if payload.get("stage_index") != stage: + raise ValueError(f"stage {stage} payload index mismatch") + if payload.get("stage_count") != expected_stages: + raise ValueError(f"stage {stage} stage-count mismatch") + if payload.get("order") != order: + raise ValueError(f"stage {stage} order mismatch") + if payload.get("formula_id") != "five_copy_suzuki_fourth_order_exact_cubic": + raise ValueError(f"stage {stage} formula mismatch") + series = payload.get("series") + if not isinstance(series, list) or len(series) != order + 1: + raise ValueError(f"stage {stage} series length mismatch") + commits.add(str(manifest.get("git_commit"))) + payloads[stage] = payload + if len(commits) != 1: + raise ValueError("shard manifests do not share one git commit") + + forward = merge_coordinate_series( + [payloads[index]["series"] for index in range(expected_stages)] + ) + reverse = merge_coordinate_series( + [payloads[index]["series"] for index in reversed(range(expected_stages))] + ) + if forward != reverse: + raise ArithmeticError("forward and reverse exact shard reductions differ") + root = cube_root_four_interval(30) + degree = forward[order] + cell_l1 = sum( + (coefficient.enclose(root).abs_upper() for coefficient in degree.values()), + Fraction(), + ) + site_l1 = cell_l1 / 4 + merged_payload = { + "schema_version": 1, + "kind": "issue128_exact_right_generator_degree", + "formula_id": "five_copy_suzuki_fourth_order_exact_cubic", + "source_commit": next(iter(commits)), + "source_stage_count": expected_stages, + "degree": order, + "terms": coordinate_terms_to_json(degree), + "cell_pauli_l1_upper": _pair(cell_l1), + "site_pauli_l1_upper": _pair(site_l1), + } + write_shard_gzip(output, merged_payload) + summary = { + "schema_version": 1, + "kind": "issue128_d8_reduction_summary", + "status": "complete", + "source_commit": next(iter(commits)), + "source_stage_count": expected_stages, + "order": order, + "degree_term_counts": { + str(index): len(terms) for index, terms in enumerate(forward) + }, + "cell_pauli_l1_upper": _pair(cell_l1), + "cell_pauli_l1_upper_decimal": _decimal(cell_l1), + "site_pauli_l1_upper": _pair(site_l1), + "site_pauli_l1_upper_decimal": _decimal(site_l1), + "output": output.name, + "output_sha256": sha256_file(output), + "reduction_order_check": "forward_equals_reverse", + } + write_manifest_atomic(summary_path, summary) + return summary + + +def main() -> None: + parser = argparse.ArgumentParser() + parser.add_argument("--input-root", type=Path, required=True) + parser.add_argument("--expected-stages", type=int, default=31) + parser.add_argument("--order", type=int, default=8) + parser.add_argument("--output", type=Path, required=True) + parser.add_argument("--summary", type=Path, required=True) + arguments = parser.parse_args() + summary = reduce_shards( + arguments.input_root, + expected_stages=arguments.expected_stages, + order=arguments.order, + output=arguments.output, + summary_path=arguments.summary, + ) + print(json.dumps(summary, indent=2), flush=True) + + +if __name__ == "__main__": + main() diff --git a/tracks/qcs/solutions/WangTheoPhys/issue128/scripts/screen_matching_order.py b/tracks/qcs/solutions/WangTheoPhys/issue128/scripts/screen_matching_order.py new file mode 100755 index 000000000..c0e7ed0e8 --- /dev/null +++ b/tracks/qcs/solutions/WangTheoPhys/issue128/scripts/screen_matching_order.py @@ -0,0 +1,69 @@ +#!/usr/bin/env python3 +from __future__ import annotations + +import argparse +from datetime import datetime, timezone +import json +import os +from pathlib import Path +import time + +from trottercert.matching_screen import screen_matching_order + + +def _write(path: Path, payload: dict[str, object]) -> None: + path.parent.mkdir(parents=True, exist_ok=True) + temporary = path.with_name(path.name + ".tmp") + temporary.write_text(json.dumps(payload, sort_keys=True, separators=(",", ":")) + "\n") + temporary.replace(path) + + +def main() -> None: + parser = argparse.ArgumentParser(description="Rank the 24 matching-color orders") + parser.add_argument("--permutation-index", type=int, required=True) + parser.add_argument("--order", type=int, default=6) + parser.add_argument("--output", type=Path, required=True) + parser.add_argument("--manifest", type=Path, required=True) + args = parser.parse_args() + started = time.perf_counter() + running = { + "schema_version": 1, + "kind": "issue128_matching_order_screen_manifest", + "status": "running", + "trusted": False, + "purpose": "discovery-ranking", + "permutation_index": args.permutation_index, + "order": args.order, + "source_commit": os.environ.get("ISSUE128_SOURCE_COMMIT", "unrecorded"), + "started_at": datetime.now(timezone.utc).isoformat(), + } + _write(args.manifest, running) + try: + payload = screen_matching_order(args.permutation_index, args.order) + _write(args.output, payload) + _write( + args.manifest, + { + **running, + "status": "complete", + "wall_seconds": time.perf_counter() - started, + "completed_at": datetime.now(timezone.utc).isoformat(), + }, + ) + except BaseException as error: + _write( + args.manifest, + { + **running, + "status": "failed", + "wall_seconds": time.perf_counter() - started, + "error_type": type(error).__name__, + "error": str(error), + "completed_at": datetime.now(timezone.utc).isoformat(), + }, + ) + raise + + +if __name__ == "__main__": + main() diff --git a/tracks/qcs/solutions/WangTheoPhys/issue128/scripts/verify.py b/tracks/qcs/solutions/WangTheoPhys/issue128/scripts/verify.py new file mode 100644 index 000000000..b30967f72 --- /dev/null +++ b/tracks/qcs/solutions/WangTheoPhys/issue128/scripts/verify.py @@ -0,0 +1,28 @@ +#!/usr/bin/env python3 +from __future__ import annotations + +import argparse +import json + +from trottercert.verify import verify_certificate + + +def main() -> None: + parser = argparse.ArgumentParser() + parser.add_argument("certificate") + parser.add_argument( + "--deep", + action="store_true", + help="regenerate every local commutator and anticommuting proof term", + ) + arguments = parser.parse_args() + print( + json.dumps( + verify_certificate(arguments.certificate, deep=arguments.deep), + indent=2, + ) + ) + + +if __name__ == "__main__": + main() diff --git a/tracks/qcs/solutions/WangTheoPhys/issue128/src/trottercert/__init__.py b/tracks/qcs/solutions/WangTheoPhys/issue128/src/trottercert/__init__.py new file mode 100644 index 000000000..9110dc0e9 --- /dev/null +++ b/tracks/qcs/solutions/WangTheoPhys/issue128/src/trottercert/__init__.py @@ -0,0 +1,5 @@ +"""Exact tools for machine-checkable Trotter error research.""" + +from .algebra import PauliString, PauliSum, QComplex, commutator + +__all__ = ["PauliString", "PauliSum", "QComplex", "commutator"] diff --git a/tracks/qcs/solutions/WangTheoPhys/issue128/src/trottercert/algebra.py b/tracks/qcs/solutions/WangTheoPhys/issue128/src/trottercert/algebra.py new file mode 100644 index 000000000..ccd44ba32 --- /dev/null +++ b/tracks/qcs/solutions/WangTheoPhys/issue128/src/trottercert/algebra.py @@ -0,0 +1,308 @@ +from __future__ import annotations + +from dataclasses import dataclass +from fractions import Fraction +from functools import reduce +from typing import Iterable, Mapping + +import numpy as np + + +def _fraction(value: int | Fraction) -> Fraction: + return value if isinstance(value, Fraction) else Fraction(value) + + +@dataclass(frozen=True, slots=True) +class QComplex: + """A complex number with exact rational real and imaginary parts.""" + + real: Fraction = Fraction(0) + imag: Fraction = Fraction(0) + + def __init__( + self, real: int | Fraction = 0, imag: int | Fraction = 0 + ) -> None: + object.__setattr__(self, "real", _fraction(real)) + object.__setattr__(self, "imag", _fraction(imag)) + + @classmethod + def coerce(cls, value: QComplex | int | Fraction) -> QComplex: + return value if isinstance(value, cls) else cls(value) + + def __add__(self, other: QComplex | int | Fraction) -> QComplex: + rhs = self.coerce(other) + return QComplex(self.real + rhs.real, self.imag + rhs.imag) + + __radd__ = __add__ + + def __neg__(self) -> QComplex: + return QComplex(-self.real, -self.imag) + + def __sub__(self, other: QComplex | int | Fraction) -> QComplex: + return self + (-self.coerce(other)) + + def __rsub__(self, other: QComplex | int | Fraction) -> QComplex: + return self.coerce(other) - self + + def __mul__(self, other: QComplex | int | Fraction) -> QComplex: + rhs = self.coerce(other) + return QComplex( + self.real * rhs.real - self.imag * rhs.imag, + self.real * rhs.imag + self.imag * rhs.real, + ) + + __rmul__ = __mul__ + + def __truediv__(self, other: int | Fraction) -> QComplex: + divisor = _fraction(other) + return QComplex(self.real / divisor, self.imag / divisor) + + def conjugate(self) -> QComplex: + return QComplex(self.real, -self.imag) + + def abs_float(self) -> float: + return float(self.real * self.real + self.imag * self.imag) ** 0.5 + + def to_complex(self) -> complex: + return complex(float(self.real), float(self.imag)) + + def is_zero(self) -> bool: + return self.real == 0 and self.imag == 0 + + +_MUL: dict[tuple[str, str], tuple[QComplex, str]] = { + ("I", "I"): (QComplex(1), "I"), + ("I", "X"): (QComplex(1), "X"), + ("I", "Y"): (QComplex(1), "Y"), + ("I", "Z"): (QComplex(1), "Z"), + ("X", "I"): (QComplex(1), "X"), + ("Y", "I"): (QComplex(1), "Y"), + ("Z", "I"): (QComplex(1), "Z"), + ("X", "X"): (QComplex(1), "I"), + ("Y", "Y"): (QComplex(1), "I"), + ("Z", "Z"): (QComplex(1), "I"), + ("X", "Y"): (QComplex(0, 1), "Z"), + ("Y", "Z"): (QComplex(0, 1), "X"), + ("Z", "X"): (QComplex(0, 1), "Y"), + ("Y", "X"): (QComplex(0, -1), "Z"), + ("Z", "Y"): (QComplex(0, -1), "X"), + ("X", "Z"): (QComplex(0, -1), "Y"), +} + + +@dataclass(frozen=True, order=True, slots=True) +class PauliString: + """A phase-free tensor-product Pauli string.""" + + ops: tuple[tuple[int, str], ...] = () + + def __init__(self, ops: Mapping[int, str] | Iterable[tuple[int, str]] = ()) -> None: + items = ops.items() if isinstance(ops, Mapping) else ops + canonical: dict[int, str] = {} + for site, op in items: + if site < 0: + raise ValueError("site indices must be nonnegative") + if op not in {"I", "X", "Y", "Z"}: + raise ValueError(f"invalid Pauli operator {op!r}") + if op != "I": + canonical[int(site)] = op + object.__setattr__(self, "ops", tuple(sorted(canonical.items()))) + + def as_dict(self) -> dict[int, str]: + return dict(self.ops) + + @property + def support(self) -> frozenset[int]: + return frozenset(site for site, _ in self.ops) + + def multiply(self, other: PauliString) -> tuple[QComplex, PauliString]: + left_index = right_index = 0 + phase_exponent = 0 + result: list[tuple[int, str]] = [] + while left_index < len(self.ops) or right_index < len(other.ops): + if right_index >= len(other.ops) or ( + left_index < len(self.ops) + and self.ops[left_index][0] < other.ops[right_index][0] + ): + result.append(self.ops[left_index]) + left_index += 1 + continue + if left_index >= len(self.ops) or ( + other.ops[right_index][0] < self.ops[left_index][0] + ): + result.append(other.ops[right_index]) + right_index += 1 + continue + site = self.ops[left_index][0] + left_op = self.ops[left_index][1] + right_op = other.ops[right_index][1] + local_phase, op = _MUL[(left_op, right_op)] + if local_phase.imag == 1: + phase_exponent += 1 + elif local_phase.imag == -1: + phase_exponent += 3 + elif local_phase.real == -1: + phase_exponent += 2 + if op != "I": + result.append((site, op)) + left_index += 1 + right_index += 1 + phases = (QComplex(1), QComplex(0, 1), QComplex(-1), QComplex(0, -1)) + return phases[phase_exponent % 4], PauliString(result) + + +class PauliSum: + """A finite exact Gaussian-rational linear combination of Pauli strings.""" + + def __init__( + self, + terms: Mapping[PauliString, QComplex | int | Fraction] | None = None, + ) -> None: + self.terms: dict[PauliString, QComplex] = {} + for pauli, coefficient in (terms or {}).items(): + self._accumulate(pauli, QComplex.coerce(coefficient)) + + @classmethod + def term( + cls, + pauli: PauliString, + coefficient: QComplex | int | Fraction = 1, + ) -> PauliSum: + return cls({pauli: coefficient}) + + @classmethod + def identity(cls, coefficient: QComplex | int | Fraction = 1) -> PauliSum: + return cls.term(PauliString(), coefficient) + + @classmethod + def zero(cls) -> PauliSum: + return cls() + + def copy(self) -> PauliSum: + return PauliSum(self.terms) + + def _accumulate(self, pauli: PauliString, coefficient: QComplex) -> None: + updated = self.terms.get(pauli, QComplex()) + coefficient + if updated.is_zero(): + self.terms.pop(pauli, None) + else: + self.terms[pauli] = updated + + def __add__(self, other: PauliSum) -> PauliSum: + result = self.copy() + for pauli, coefficient in other.terms.items(): + result._accumulate(pauli, coefficient) + return result + + def __sub__(self, other: PauliSum) -> PauliSum: + return self + (-other) + + def __neg__(self) -> PauliSum: + return self.scale(-1) + + def scale(self, coefficient: QComplex | int | Fraction) -> PauliSum: + scalar = QComplex.coerce(coefficient) + return PauliSum({p: scalar * c for p, c in self.terms.items()}) + + def __mul__(self, other: PauliSum | QComplex | int | Fraction) -> PauliSum: + if not isinstance(other, PauliSum): + return self.scale(other) + result = PauliSum.zero() + for left, left_coefficient in self.terms.items(): + for right, right_coefficient in other.terms.items(): + phase, pauli = left.multiply(right) + result._accumulate(pauli, left_coefficient * right_coefficient * phase) + return result + + def __rmul__(self, other: QComplex | int | Fraction) -> PauliSum: + return self.scale(other) + + def dagger(self) -> PauliSum: + return PauliSum({p: c.conjugate() for p, c in self.terms.items()}) + + def is_hermitian(self) -> bool: + return self.terms == self.dagger().terms + + def pauli_l1(self) -> float: + return sum(coefficient.abs_float() for coefficient in self.terms.values()) + + def exact_real_l1(self) -> Fraction: + total = Fraction() + for coefficient in self.terms.values(): + if coefficient.imag != 0: + raise ValueError("exact_real_l1 requires real coefficients") + total += abs(coefficient.real) + return total + + def exact_axis_l1(self) -> Fraction: + """Exact l1 norm when every coefficient is purely real or imaginary.""" + + total = Fraction() + for coefficient in self.terms.values(): + if coefficient.real and coefficient.imag: + raise ValueError("coefficient modulus is irrational in general") + total += abs(coefficient.real) + abs(coefficient.imag) + return total + + def __eq__(self, other: object) -> bool: + return isinstance(other, PauliSum) and self.terms == other.terms + + def __bool__(self) -> bool: + return bool(self.terms) + + +def commutator(left: PauliSum, right: PauliSum) -> PauliSum: + result = PauliSum.zero() + for left_pauli, left_coefficient in left.terms.items(): + for right_pauli, right_coefficient in right.terms.items(): + if pauli_strings_commute(left_pauli, right_pauli): + continue + phase, product = left_pauli.multiply(right_pauli) + result._accumulate( + product, + QComplex(2) * left_coefficient * right_coefficient * phase, + ) + return result + + +def pauli_strings_commute(left: PauliString, right: PauliString) -> bool: + left_index = right_index = 0 + parity = 0 + while left_index < len(left.ops) and right_index < len(right.ops): + left_site, left_op = left.ops[left_index] + right_site, right_op = right.ops[right_index] + if left_site < right_site: + left_index += 1 + elif right_site < left_site: + right_index += 1 + else: + if left_op != right_op: + parity ^= 1 + left_index += 1 + right_index += 1 + if parity and ( + left_index >= len(left.ops) or right_index >= len(right.ops) + ): + break + return parity == 0 + + +_DENSE = { + "I": np.eye(2, dtype=complex), + "X": np.array([[0, 1], [1, 0]], dtype=complex), + "Y": np.array([[0, -1j], [1j, 0]], dtype=complex), + "Z": np.array([[1, 0], [0, -1]], dtype=complex), +} + + +def to_dense(operator: PauliSum, n_qubits: int) -> np.ndarray: + if n_qubits < 0: + raise ValueError("n_qubits must be nonnegative") + dimension = 1 << n_qubits + result = np.zeros((dimension, dimension), dtype=complex) + for pauli, coefficient in operator.terms.items(): + local = pauli.as_dict() + factors = [_DENSE[local.get(site, "I")] for site in range(n_qubits)] + matrix = reduce(np.kron, factors, np.array([[1]], dtype=complex)) + result += coefficient.to_complex() * matrix + return result diff --git a/tracks/qcs/solutions/WangTheoPhys/issue128/src/trottercert/anticommuting.py b/tracks/qcs/solutions/WangTheoPhys/issue128/src/trottercert/anticommuting.py new file mode 100644 index 000000000..d2053a66f --- /dev/null +++ b/tracks/qcs/solutions/WangTheoPhys/issue128/src/trottercert/anticommuting.py @@ -0,0 +1,149 @@ +from __future__ import annotations + +from collections.abc import Mapping, Sequence +from dataclasses import dataclass +from fractions import Fraction +from math import isqrt + +from .intervals import RationalInterval +from .local_commutators import SymplecticPauli + + +@dataclass(frozen=True) +class AnticommutingGroupCertificate: + term_indices: tuple[int, ...] + bound: Fraction + + +@dataclass(frozen=True) +class AnticommutingPartitionCertificate: + paulis: tuple[SymplecticPauli, ...] + coefficients: tuple[RationalInterval, ...] + groups: tuple[AnticommutingGroupCertificate, ...] + bound: Fraction + + +def symplectic_anticommutes( + left: SymplecticPauli, + right: SymplecticPauli, +) -> bool: + left_x, left_z = left + right_x, right_z = right + return bool( + ( + (left_x & right_z).bit_count() + + (left_z & right_x).bit_count() + ) + & 1 + ) + + +def sqrt_fraction_upper( + value: Fraction, + *, + decimal_places: int = 30, +) -> Fraction: + if value < 0: + raise ValueError("cannot bound the square root of a negative value") + if decimal_places < 0: + raise ValueError("decimal places must be nonnegative") + if value == 0: + return Fraction() + scale = 10**decimal_places + quotient = value.numerator * scale * scale // value.denominator + root = isqrt(quotient) + candidate = Fraction(root, scale) + if candidate * candidate < value: + candidate = Fraction(root + 1, scale) + if candidate * candidate < value: + raise ArithmeticError("outward square-root rounding failed") + return candidate + + +def discover_anticommuting_partition( + coefficients: Mapping[SymplecticPauli, RationalInterval], + *, + max_group_size: int = 10, +) -> tuple[tuple[SymplecticPauli, ...], ...]: + if max_group_size < 1: + raise ValueError("maximum group size must be positive") + ordered = tuple( + sorted( + coefficients, + key=lambda pauli: ( + -float(coefficients[pauli].abs_upper()), + pauli, + ), + ) + ) + used: set[SymplecticPauli] = set() + groups: list[tuple[SymplecticPauli, ...]] = [] + for position, pauli in enumerate(ordered): + if pauli in used: + continue + group = [pauli] + used.add(pauli) + for candidate in ordered[position + 1 :]: + if candidate in used: + continue + if all( + symplectic_anticommutes(candidate, member) + for member in group + ): + group.append(candidate) + used.add(candidate) + if len(group) == max_group_size: + break + groups.append(tuple(group)) + return tuple(groups) + + +def certify_anticommuting_partition( + coefficients: Mapping[SymplecticPauli, RationalInterval], + groups: Sequence[Sequence[SymplecticPauli]], + *, + sqrt_decimal_places: int = 30, +) -> AnticommutingPartitionCertificate: + paulis = tuple(sorted(coefficients)) + index = {pauli: offset for offset, pauli in enumerate(paulis)} + flattened = tuple(pauli for group in groups for pauli in group) + if len(flattened) != len(set(flattened)): + raise ValueError("partition coverage contains duplicate terms") + if set(flattened) != set(paulis): + raise ValueError("partition coverage differs from coefficient map") + + certified_groups: list[AnticommutingGroupCertificate] = [] + for group in groups: + if not group: + raise ValueError("anticommuting group must be nonempty") + for left_position, left in enumerate(group): + for right in group[left_position + 1 :]: + if not symplectic_anticommutes(left, right): + raise ValueError("group members do not anticommute") + squared = sum( + ( + coefficients[pauli].abs_upper() ** 2 + for pauli in group + ), + Fraction(), + ) + bound = sqrt_fraction_upper( + squared, + decimal_places=sqrt_decimal_places, + ) + certified_groups.append( + AnticommutingGroupCertificate( + tuple(index[pauli] for pauli in group), + bound, + ) + ) + + return AnticommutingPartitionCertificate( + paulis=paulis, + coefficients=tuple(coefficients[pauli] for pauli in paulis), + groups=tuple(certified_groups), + bound=sum( + (group.bound for group in certified_groups), + Fraction(), + ), + ) diff --git a/tracks/qcs/solutions/WangTheoPhys/issue128/src/trottercert/baseline.py b/tracks/qcs/solutions/WangTheoPhys/issue128/src/trottercert/baseline.py new file mode 100644 index 000000000..e5b1923e2 --- /dev/null +++ b/tracks/qcs/solutions/WangTheoPhys/issue128/src/trottercert/baseline.py @@ -0,0 +1,48 @@ +from __future__ import annotations + +from dataclasses import dataclass +from fractions import Fraction +from typing import Callable, Sequence + +from .algebra import PauliSum, commutator + + +@dataclass(frozen=True) +class StrangCommutatorBlock: + fragment_index: int + repeated_fragment: PauliSum + repeated_tail: PauliSum + + +def _sum_ops(operators: Sequence[PauliSum]) -> PauliSum: + result = PauliSum.zero() + for operator in operators: + result += operator + return result + + +def strang_commutator_operators( + fragments: Sequence[PauliSum], +) -> tuple[StrangCommutatorBlock, ...]: + if len(fragments) < 2: + raise ValueError("Strang formula requires at least two fragments") + blocks: list[StrangCommutatorBlock] = [] + for index in range(len(fragments) - 1): + current = fragments[index] + tail = _sum_ops(fragments[index + 1 :]) + blocks.append( + StrangCommutatorBlock( + fragment_index=index, + repeated_fragment=commutator(current, commutator(current, tail)), + repeated_tail=commutator(tail, commutator(tail, current)), + ) + ) + return tuple(blocks) + + +def pauli_l1_second_order_constant(fragments: Sequence[PauliSum]) -> Fraction: + total = Fraction() + for block in strang_commutator_operators(fragments): + total += Fraction(1, 24) * block.repeated_fragment.exact_real_l1() + total += Fraction(1, 12) * block.repeated_tail.exact_real_l1() + return total diff --git a/tracks/qcs/solutions/WangTheoPhys/issue128/src/trottercert/clusters.py b/tracks/qcs/solutions/WangTheoPhys/issue128/src/trottercert/clusters.py new file mode 100644 index 000000000..72a49a61b --- /dev/null +++ b/tracks/qcs/solutions/WangTheoPhys/issue128/src/trottercert/clusters.py @@ -0,0 +1,520 @@ +from __future__ import annotations + +from dataclasses import dataclass +from fractions import Fraction +from math import isqrt +from math import lcm + +import numpy as np + +from .algebra import PauliString, PauliSum, QComplex, pauli_strings_commute +from .orbits import TranslationOrbit, embed_grid_op +from .lattice import SquareLattice +from .orbits import translation_orbits + + +@dataclass(frozen=True) +class ClusterPlacement: + orbit_index: int + offsets: tuple[tuple[int, int], ...] + weights: tuple[Fraction, ...] + + def validate(self) -> None: + if not self.offsets or len(self.offsets) != len(self.weights): + raise ValueError("placement offsets and weights must be nonempty and aligned") + if any(weight < 0 for weight in self.weights): + raise ValueError("cluster weights must be nonnegative") + if sum(self.weights, Fraction()) != 1: + raise ValueError("weights for each translation orbit must sum to one") + + +@dataclass(frozen=True) +class ClusterCertificate: + width: int + height: int + placements: tuple[ClusterPlacement, ...] + operator: PauliSum + row_sum_bound: Fraction + + +@dataclass(frozen=True) +class PartitionedOperatorCertificate: + unit_cell: tuple[int, int] + n_cells: int + horizontal: ClusterCertificate + vertical: ClusterCertificate + global_bound: Fraction + + +@dataclass(frozen=True) +class FullPatchOperatorCertificate: + unit_cell: tuple[int, int] + n_cells: int + cluster_operator: PauliSum + placements: tuple[ClusterPlacement, ...] + density_bound: Fraction + global_bound: Fraction + + +@dataclass(frozen=True) +class CollatzCertificate: + denominator: int + vector: tuple[int, ...] + max_numerator: int + max_denominator_component: int + bound: Fraction + + +@dataclass(frozen=True) +class AnticommutingCertificate: + groups: tuple[tuple[PauliString, ...], ...] + bound: Fraction + + +@dataclass(frozen=True) +class PhasePartitionedCertificate: + unit_cell: tuple[int, int] + n_cells: int + phase_bounds: tuple[tuple[tuple[int, int], CollatzCertificate], ...] + global_bound: Fraction + + +def uniform_cluster_operator( + orbits: tuple[TranslationOrbit, ...], + width: int, + height: int, +) -> tuple[PauliSum, tuple[ClusterPlacement, ...]]: + cluster = PauliSum.zero() + placements: list[ClusterPlacement] = [] + for orbit_index, orbit in enumerate(orbits): + if orbit.width > width or orbit.height > height: + raise ValueError( + f"orbit {orbit_index} with {orbit.width}x{orbit.height} support " + f"does not fit {width}x{height} patch" + ) + step_x, step_y = orbit.unit_cell + offsets = tuple( + (dx, dy) + for dy in range(0, height - orbit.height + 1, step_y) + for dx in range(0, width - orbit.width + 1, step_x) + ) + if not offsets: + raise ValueError(f"orbit {orbit_index} has no unit-cell-aligned placement") + weight = Fraction(1, len(offsets)) + weights = tuple(weight for _ in offsets) + placement = ClusterPlacement(orbit_index, offsets, weights) + placement.validate() + placements.append(placement) + for (dx, dy), local_weight in zip(offsets, weights): + cluster += PauliSum.term( + embed_grid_op(orbit.representative, width, dx, dy), + orbit.coefficient * local_weight, + ) + return cluster, tuple(placements) + + +def _sqrt_fraction_upper(value: Fraction, decimal_places: int = 30) -> Fraction: + if value < 0: + raise ValueError("cannot bound square root of negative value") + if value == 0: + return Fraction() + scale = 10**decimal_places + quotient = (value.numerator * scale * scale) // value.denominator + root = isqrt(quotient) + if Fraction(root * root, scale * scale) < value: + root += 1 + return Fraction(root, scale) + + +def paulis_anticommute(left: PauliString, right: PauliString) -> bool: + return not pauli_strings_commute(left, right) + + +def greedy_anticommuting_certificate(operator: PauliSum) -> AnticommutingCertificate: + if any(coefficient.imag for coefficient in operator.terms.values()): + raise ValueError("anticommuting certificate currently requires real coefficients") + ordered = sorted( + operator.terms, + key=lambda pauli: ( + -abs(operator.terms[pauli].real), + pauli, + ), + ) + masks: dict[PauliString, tuple[int, int]] = {} + for pauli in ordered: + x_mask = z_mask = 0 + for site, op in pauli.ops: + bit = 1 << site + if op in {"X", "Y"}: + x_mask |= bit + if op in {"Z", "Y"}: + z_mask |= bit + masks[pauli] = (x_mask, z_mask) + + def fast_anticommutes(left: PauliString, right: PauliString) -> bool: + left_x, left_z = masks[left] + right_x, right_z = masks[right] + return bool( + ((left_x & right_z).bit_count() + (left_z & right_x).bit_count()) + & 1 + ) + + groups: list[list[PauliString]] = [] + for pauli in ordered: + compatible = [ + index + for index, group in enumerate(groups) + if all(fast_anticommutes(pauli, member) for member in group) + ] + if compatible: + # Prefer the largest compatible group to reduce the number of + # triangle inequalities deterministically. + selected = max(compatible, key=lambda index: (len(groups[index]), -index)) + groups[selected].append(pauli) + else: + groups.append([pauli]) + bound = Fraction() + for group in groups: + squared = sum( + (operator.terms[pauli].real ** 2 for pauli in group), + Fraction(), + ) + bound += _sqrt_fraction_upper(squared) + return AnticommutingCertificate( + groups=tuple(tuple(group) for group in groups), + bound=bound, + ) + + +def computational_row_sum_bound(operator: PauliSum, n_qubits: int) -> Fraction: + """Rigorous max absolute row-sum bound using exact merged amplitudes.""" + + if n_qubits > 16: + raise ValueError("row enumeration is restricted to at most 16 qubits") + best = Fraction() + parsed = [] + for pauli, coefficient in operator.terms.items(): + flip_mask = 0 + z_sites: list[int] = [] + y_sites: list[int] = [] + for site, op in pauli.ops: + if op in {"X", "Y"}: + flip_mask |= 1 << (n_qubits - 1 - site) + if op == "Z": + z_sites.append(site) + elif op == "Y": + y_sites.append(site) + parsed.append((flip_mask, tuple(z_sites), tuple(y_sites), coefficient)) + + for basis in range(1 << n_qubits): + amplitudes: dict[int, QComplex] = {} + for flip_mask, z_sites, y_sites, coefficient in parsed: + phase = QComplex(1) + for site in z_sites: + bit = (basis >> (n_qubits - 1 - site)) & 1 + if bit: + phase = -phase + for site in y_sites: + bit = (basis >> (n_qubits - 1 - site)) & 1 + phase *= QComplex(0, -1 if bit else 1) + target = basis ^ flip_mask + amplitudes[target] = amplitudes.get(target, QComplex()) + coefficient * phase + row = Fraction() + for amplitude in amplitudes.values(): + squared = amplitude.real**2 + amplitude.imag**2 + row += _sqrt_fraction_upper(squared) + best = max(best, row) + return best + + +def computational_row_taxicab_bound(operator: PauliSum, n_qubits: int) -> Fraction: + """Fast exact row bound using ``|z| <= |Re z| + |Im z|``. + + Coefficients are lifted to a common integer denominator. Amplitudes with + the same computational-basis transition are merged exactly using int64 + arrays, retaining cancellations before the taxicab inequality. + """ + + if n_qubits > 20: + raise ValueError("taxicab row enumeration is restricted to at most 20 qubits") + denominator = 1 + for coefficient in operator.terms.values(): + denominator = lcm( + denominator, + coefficient.real.denominator, + coefficient.imag.denominator, + ) + grouped: dict[int, list[tuple[int, int, int, int]]] = {} + max_scaled = 0 + for pauli, coefficient in operator.terms.items(): + flip_mask = 0 + phase_mask = 0 + n_y = 0 + for site, op in pauli.ops: + bit_mask = 1 << (n_qubits - 1 - site) + if op in {"X", "Y"}: + flip_mask |= bit_mask + if op in {"Z", "Y"}: + phase_mask |= bit_mask + if op == "Y": + n_y += 1 + real = coefficient.real.numerator * ( + denominator // coefficient.real.denominator + ) + imag = coefficient.imag.numerator * ( + denominator // coefficient.imag.denominator + ) + phase = n_y % 4 + if phase == 1: + real, imag = -imag, real + elif phase == 2: + real, imag = -real, -imag + elif phase == 3: + real, imag = imag, -real + max_scaled += abs(real) + abs(imag) + grouped.setdefault(flip_mask, []).append((phase_mask, real, imag, n_y)) + if max_scaled >= np.iinfo(np.int64).max // max(1, len(operator.terms)): + raise OverflowError("scaled exact amplitudes exceed int64 safety margin") + + size = 1 << n_qubits + indices = np.arange(size, dtype=np.uint64) + parity = np.fromiter( + ((index.bit_count() & 1) for index in range(size)), + dtype=np.int8, + count=size, + ) + row_sums = np.zeros(size, dtype=np.int64) + for terms in grouped.values(): + real_amplitude = np.zeros(size, dtype=np.int64) + imag_amplitude = np.zeros(size, dtype=np.int64) + for phase_mask, real, imag, _ in terms: + signs = 1 - 2 * parity[np.bitwise_and(indices, phase_mask)] + real_amplitude += real * signs + imag_amplitude += imag * signs + row_sums += np.abs(real_amplitude) + np.abs(imag_amplitude) + return Fraction(int(row_sums.max()), denominator) + + +def _taxicab_transition_groups( + operator: PauliSum, + n_qubits: int, +) -> tuple[int, tuple[tuple[int, np.ndarray], ...]]: + denominator = 1 + for coefficient in operator.terms.values(): + denominator = lcm( + denominator, + coefficient.real.denominator, + coefficient.imag.denominator, + ) + grouped: dict[int, list[tuple[int, int, int]]] = {} + for pauli, coefficient in operator.terms.items(): + flip_mask = 0 + phase_mask = 0 + n_y = 0 + for site, op in pauli.ops: + bit_mask = 1 << (n_qubits - 1 - site) + if op in {"X", "Y"}: + flip_mask |= bit_mask + if op in {"Z", "Y"}: + phase_mask |= bit_mask + if op == "Y": + n_y += 1 + real = coefficient.real.numerator * ( + denominator // coefficient.real.denominator + ) + imag = coefficient.imag.numerator * ( + denominator // coefficient.imag.denominator + ) + phase = n_y % 4 + if phase == 1: + real, imag = -imag, real + elif phase == 2: + real, imag = -real, -imag + elif phase == 3: + real, imag = imag, -real + grouped.setdefault(flip_mask, []).append((phase_mask, real, imag)) + + size = 1 << n_qubits + indices = np.arange(size, dtype=np.uint64) + parity = np.fromiter( + ((index.bit_count() & 1) for index in range(size)), + dtype=np.int8, + count=size, + ) + transitions: list[tuple[int, np.ndarray]] = [] + for flip_mask, terms in grouped.items(): + real_amplitude = np.zeros(size, dtype=np.int64) + imag_amplitude = np.zeros(size, dtype=np.int64) + for phase_mask, real, imag in terms: + signs = 1 - 2 * parity[np.bitwise_and(indices, phase_mask)] + real_amplitude += real * signs + imag_amplitude += imag * signs + values = np.abs(real_amplitude) + np.abs(imag_amplitude) + transitions.append((flip_mask, values)) + return denominator, tuple(transitions) + + +def collatz_taxicab_certificate( + operator: PauliSum, + n_qubits: int, + *, + iterations: int = 40, + quantization: int = 10**9, +) -> CollatzCertificate: + """Certify the taxicab comparison-matrix norm by Collatz--Wielandt.""" + + denominator, transitions = _taxicab_transition_groups(operator, n_qubits) + size = 1 << n_qubits + indices = np.arange(size, dtype=np.uint64) + vector = np.ones(size, dtype=np.float64) + for _ in range(iterations): + updated = np.zeros(size, dtype=np.float64) + for flip_mask, values in transitions: + updated += values * vector[np.bitwise_xor(indices, flip_mask)] + maximum = float(updated.max()) + if maximum == 0: + return CollatzCertificate( + denominator, tuple([1] * size), 0, 1, Fraction() + ) + vector = updated / maximum + vector += np.finfo(np.float64).eps + integer_vector = np.maximum(1, np.rint(vector * quantization)).astype(np.int64) + + applied = np.zeros(size, dtype=np.int64) + for flip_mask, values in transitions: + contribution = values * integer_vector[np.bitwise_xor(indices, flip_mask)] + if np.any(contribution < 0): + raise OverflowError("integer overflow in Collatz verification") + applied += contribution + if np.any(applied < 0): + raise OverflowError("integer overflow in Collatz verification") + + best_index = max( + range(size), + key=lambda index: Fraction(int(applied[index]), int(integer_vector[index])), + ) + numerator = int(applied[best_index]) + vector_component = int(integer_vector[best_index]) + bound = Fraction(numerator, denominator * vector_component) + return CollatzCertificate( + denominator=denominator, + vector=tuple(int(value) for value in integer_vector), + max_numerator=numerator, + max_denominator_component=vector_component, + bound=bound, + ) + + +def phase_partitioned_collatz_certificate( + operator: PauliSum, + lattice: SquareLattice, + unit_cell: tuple[int, int] = (2, 2), + *, + iterations: int = 35, +) -> PhasePartitionedCertificate: + """Use separate patch origins for distinct colored unit-cell phases.""" + + orbits = translation_orbits(operator, lattice, unit_cell=unit_cell) + grouped: dict[tuple[int, int], list[TranslationOrbit]] = {} + for orbit in orbits: + min_x = min(x for x, _, _ in orbit.representative) + min_y = min(y for _, y, _ in orbit.representative) + phase = (min_x % unit_cell[0], min_y % unit_cell[1]) + normalized = tuple( + (x - phase[0], y - phase[1], op) + for x, y, op in orbit.representative + ) + grouped.setdefault(phase, []).append( + TranslationOrbit( + normalized, + orbit.coefficient, + orbit.members, + orbit.unit_cell, + orbit.global_coefficient, + orbit.stabilizer, + ) + ) + phase_bounds: list[tuple[tuple[int, int], CollatzCertificate]] = [] + density_sum = Fraction() + for phase in sorted(grouped): + cluster, _ = uniform_cluster_operator(tuple(grouped[phase]), 4, 4) + certificate = collatz_taxicab_certificate( + cluster, 16, iterations=iterations + ) + phase_bounds.append((phase, certificate)) + density_sum += certificate.bound + step_x, step_y = unit_cell + n_cells = (lattice.length // step_x) * (lattice.length // step_y) + return PhasePartitionedCertificate( + unit_cell, + n_cells, + tuple(phase_bounds), + n_cells * density_sum, + ) + + +def build_uniform_cluster_certificate( + orbits: tuple[TranslationOrbit, ...], + width: int, + height: int, +) -> ClusterCertificate: + operator, placements = uniform_cluster_operator(orbits, width, height) + bound = computational_row_sum_bound(operator, width * height) + return ClusterCertificate(width, height, placements, operator, bound) + + +def build_partitioned_operator_certificate( + operator: PauliSum, + lattice: SquareLattice, + unit_cell: tuple[int, int] = (2, 2), +) -> PartitionedOperatorCertificate: + """Certify a colored translation-invariant range-three operator. + + Horizontal/diagonal-support representatives with ``width >= height`` use + a 4x3 patch; the others use a 3x4 patch. Splitting costs one triangle + inequality but keeps each exact row enumeration at 12 qubits. + """ + + orbits = translation_orbits(operator, lattice, unit_cell=unit_cell) + horizontal_orbits = tuple( + orbit for orbit in orbits if orbit.width >= orbit.height + ) + vertical_orbits = tuple( + orbit for orbit in orbits if orbit.width < orbit.height + ) + horizontal = build_uniform_cluster_certificate(horizontal_orbits, 4, 3) + vertical = build_uniform_cluster_certificate(vertical_orbits, 3, 4) + step_x, step_y = unit_cell + n_cells = (lattice.length // step_x) * (lattice.length // step_y) + global_bound = n_cells * ( + horizontal.row_sum_bound + vertical.row_sum_bound + ) + return PartitionedOperatorCertificate( + unit_cell, + n_cells, + horizontal, + vertical, + global_bound, + ) + + +def build_full_patch_operator_certificate( + operator: PauliSum, + lattice: SquareLattice, + unit_cell: tuple[int, int] = (2, 2), + patch: tuple[int, int] = (4, 4), +) -> FullPatchOperatorCertificate: + orbits = translation_orbits(operator, lattice, unit_cell=unit_cell) + width, height = patch + cluster, placements = uniform_cluster_operator(orbits, width, height) + density_bound = computational_row_taxicab_bound(cluster, width * height) + step_x, step_y = unit_cell + n_cells = (lattice.length // step_x) * (lattice.length // step_y) + return FullPatchOperatorCertificate( + unit_cell, + n_cells, + cluster, + placements, + density_bound, + n_cells * density_bound, + ) diff --git a/tracks/qcs/solutions/WangTheoPhys/issue128/src/trottercert/crosscheck.py b/tracks/qcs/solutions/WangTheoPhys/issue128/src/trottercert/crosscheck.py new file mode 100644 index 000000000..6bd05747c --- /dev/null +++ b/tracks/qcs/solutions/WangTheoPhys/issue128/src/trottercert/crosscheck.py @@ -0,0 +1,62 @@ +from __future__ import annotations + +from fractions import Fraction + +import numpy as np +from scipy.linalg import expm + +from .algebra import to_dense +from .hamiltonian import four_matching_fragments, fragment_from_bonds +from .higher_order import fourth_order_suzuki_stages +from .lattice import SquareLattice +# A short outward-rounded summary of the schema-v3 bound at N=144, r=97. +# The exact rational value remains in certificates/issue128-certificate.json. +REFINED_BOUND_N144 = Fraction(996, 10**9) +REFINED_STEPS = 97 + + +def small_exact_crosscheck( + length: int, + tolerance: Fraction, +) -> dict[str, object]: + if length == 2: + groups = ( + ((0, 1), (2, 3)), + ((0, 1), (2, 3)), + ((0, 2), (1, 3)), + ((0, 2), (1, 3)), + ) + fragments = tuple(fragment_from_bonds(group) for group in groups) + n_sites = 4 + else: + lattice = SquareLattice(length) + fragments = four_matching_fragments(lattice) + n_sites = lattice.n_sites + matrices = [to_dense(fragment, n_sites) for fragment in fragments] + hamiltonian = sum(matrices, np.zeros_like(matrices[0])) + steps = REFINED_STEPS + delta = 1 / steps + one_step = np.eye(1 << n_sites, dtype=np.complex128) + for stage in fourth_order_suzuki_stages(4): + one_step = one_step @ expm( + -1j * float(stage.coefficient) * delta * matrices[stage.fragment_index] + ) + approximation = np.linalg.matrix_power(one_step, steps) + exact = expm(-1j * hamiltonian) + error = float(np.linalg.norm(approximation - exact, ord=2)) + bound = float(REFINED_BOUND_N144 * Fraction(n_sites, 144)) + return { + "length": length, + "n_sites": n_sites, + "model_note": ( + "degenerate 2x2 periodic algebra sanity check for schema-v3 r=97" + if length == 2 + else "periodic square lattice" + ), + "steps": steps, + "empirical_operator_norm_error": error, + "certified_upper_bound": bound, + "requested_tolerance": float(tolerance), + "bound_dominates_empirical_error": bound >= error, + "bound_meets_tolerance": bound <= float(tolerance), + } diff --git a/tracks/qcs/solutions/WangTheoPhys/issue128/src/trottercert/cubic_field.py b/tracks/qcs/solutions/WangTheoPhys/issue128/src/trottercert/cubic_field.py new file mode 100644 index 000000000..c4c7f1154 --- /dev/null +++ b/tracks/qcs/solutions/WangTheoPhys/issue128/src/trottercert/cubic_field.py @@ -0,0 +1,163 @@ +from __future__ import annotations + +from dataclasses import dataclass +from fractions import Fraction +from typing import Sequence + +from .intervals import RationalInterval + + +Rational = int | Fraction + + +@dataclass(frozen=True, slots=True) +class Cubic: + """An exact element of ``Q[alpha] / (alpha**3 - 4)``.""" + + a0: Fraction + a1: Fraction + a2: Fraction + + def __init__(self, a0: Rational, a1: Rational, a2: Rational) -> None: + object.__setattr__(self, "a0", Fraction(a0)) + object.__setattr__(self, "a1", Fraction(a1)) + object.__setattr__(self, "a2", Fraction(a2)) + + @classmethod + def zero(cls) -> Cubic: + return cls(0, 0, 0) + + @classmethod + def one(cls) -> Cubic: + return cls(1, 0, 0) + + @classmethod + def coerce(cls, value: Cubic | Rational) -> Cubic: + if isinstance(value, cls): + return value + if isinstance(value, (int, Fraction)): + return cls(value, 0, 0) + raise TypeError(f"cannot coerce {type(value).__name__} to Cubic") + + def __add__(self, other: Cubic | Rational) -> Cubic: + rhs = self.coerce(other) + return Cubic( + self.a0 + rhs.a0, + self.a1 + rhs.a1, + self.a2 + rhs.a2, + ) + + __radd__ = __add__ + + def __neg__(self) -> Cubic: + return Cubic(-self.a0, -self.a1, -self.a2) + + def __sub__(self, other: Cubic | Rational) -> Cubic: + return self + (-self.coerce(other)) + + def __rsub__(self, other: Cubic | Rational) -> Cubic: + return self.coerce(other) - self + + def __mul__(self, other: Cubic | Rational) -> Cubic: + rhs = self.coerce(other) + return Cubic( + self.a0 * rhs.a0 + + 4 * (self.a1 * rhs.a2 + self.a2 * rhs.a1), + self.a0 * rhs.a1 + + self.a1 * rhs.a0 + + 4 * self.a2 * rhs.a2, + self.a0 * rhs.a2 + self.a1 * rhs.a1 + self.a2 * rhs.a0, + ) + + __rmul__ = __mul__ + + def __truediv__(self, other: Rational) -> Cubic: + denominator = Fraction(other) + if denominator == 0: + raise ZeroDivisionError("division by zero") + return Cubic( + self.a0 / denominator, + self.a1 / denominator, + self.a2 / denominator, + ) + + def __pow__(self, exponent: int) -> Cubic: + if not isinstance(exponent, int): + raise TypeError("Cubic exponent must be an integer") + if exponent < 0: + raise ValueError("negative Cubic powers are not implemented") + result = self.one() + base = self + power = exponent + while power: + if power & 1: + result = result * base + base = base * base + power >>= 1 + return result + + def enclose(self, root: RationalInterval) -> RationalInterval: + """Enclose this element when ``alpha`` lies in ``root``.""" + + return ( + RationalInterval.point(self.a0) + + self.a1 * root + + self.a2 * root**2 + ) + + +@dataclass(frozen=True, slots=True) +class CubicStage: + fragment_index: int + coefficient: Cubic + + +def _second_order_cubic_stages( + n_fragments: int, + scale: Cubic, +) -> list[CubicStage]: + half = scale / 2 + stages = [ + CubicStage(fragment_index, half) + for fragment_index in range(n_fragments - 1) + ] + stages.append(CubicStage(n_fragments - 1, scale)) + stages.extend( + CubicStage(fragment_index, half) + for fragment_index in reversed(range(n_fragments - 1)) + ) + return stages + + +def _merge_cubic_stages( + stages: Sequence[CubicStage], +) -> tuple[CubicStage, ...]: + merged: list[CubicStage] = [] + for stage in stages: + if merged and merged[-1].fragment_index == stage.fragment_index: + previous = merged.pop() + merged.append( + CubicStage( + stage.fragment_index, + previous.coefficient + stage.coefficient, + ) + ) + else: + merged.append(stage) + return tuple(merged) + + +def fourth_order_suzuki_cubic_stages( + n_fragments: int = 4, +) -> tuple[CubicStage, ...]: + """Return the fourth-order Suzuki stages with exact algebraic weights.""" + + if n_fragments < 1: + raise ValueError("number of fragments must be positive") + # alpha**3 = 4 and 1 / (4 - alpha) = (16 + 4 alpha + alpha**2) / 60. + u = Cubic(Fraction(4, 15), Fraction(1, 15), Fraction(1, 60)) + scales = (u, u, Cubic.one() - 4 * u, u, u) + stages: list[CubicStage] = [] + for scale in scales: + stages.extend(_second_order_cubic_stages(n_fragments, scale)) + return _merge_cubic_stages(stages) diff --git a/tracks/qcs/solutions/WangTheoPhys/issue128/src/trottercert/cubic_local.py b/tracks/qcs/solutions/WangTheoPhys/issue128/src/trottercert/cubic_local.py new file mode 100644 index 000000000..be6d7821d --- /dev/null +++ b/tracks/qcs/solutions/WangTheoPhys/issue128/src/trottercert/cubic_local.py @@ -0,0 +1,302 @@ +from __future__ import annotations + +from fractions import Fraction +from math import factorial +from typing import Mapping, Sequence + +from .cubic_field import Cubic, CubicStage +from .local_commutators import ( + CoordinateBond, + CoordinateRegistry, + SymplecticPauli, + _iter_set_bits, + _symplectic_anticommutes, + _symplectic_bond_terms, + _symplectic_product_phase, + canonical_coordinate_bond, + matching_partner, + SymplecticDyadicLocalDensityEvaluator, +) +from .refined_error import canonicalize_symplectic_unit_cell + + +CubicTerms = dict[SymplecticPauli, Cubic] +CubicWord = tuple[int, ...] +CubicWordSeries = list[dict[CubicWord, Cubic]] + + +def _add_term( + target: dict[object, Cubic], + key: object, + value: Cubic, +) -> None: + if value == Cubic.zero(): + return + updated = target.get(key, Cubic.zero()) + value + if updated == Cubic.zero(): + target.pop(key, None) + else: + target[key] = updated + + +def _series_identity(order: int) -> CubicWordSeries: + result: CubicWordSeries = [{} for _ in range(order + 1)] + result[0][()] = Cubic.one() + return result + + +def _series_add( + left: CubicWordSeries, + right: CubicWordSeries, +) -> CubicWordSeries: + result: CubicWordSeries = [] + for left_degree, right_degree in zip(left, right): + terms = dict(left_degree) + for word, coefficient in right_degree.items(): + _add_term(terms, word, coefficient) + result.append(terms) + return result + + +def _series_scale( + series: CubicWordSeries, + scalar: int | Fraction, +) -> CubicWordSeries: + return [ + { + word: scaled + for word, coefficient in degree.items() + if (scaled := coefficient * scalar) != Cubic.zero() + } + for degree in series + ] + + +def _series_multiply( + left: CubicWordSeries, + right: CubicWordSeries, +) -> CubicWordSeries: + order = len(left) - 1 + result: CubicWordSeries = [{} for _ in range(order + 1)] + for degree in range(order + 1): + terms = result[degree] + for left_degree in range(degree + 1): + for left_word, left_coefficient in left[left_degree].items(): + for right_word, right_coefficient in right[ + degree - left_degree + ].items(): + _add_term( + terms, + left_word + right_word, + left_coefficient * right_coefficient, + ) + return result + + +def _stage_exponential( + stage: CubicStage, + order: int, +) -> CubicWordSeries: + result: CubicWordSeries = [{} for _ in range(order + 1)] + for degree in range(order + 1): + result[degree][(stage.fragment_index,) * degree] = ( + stage.coefficient**degree / factorial(degree) + ) + return result + + +def cubic_formula_log_series( + stages: Sequence[CubicStage], + order: int, +) -> CubicWordSeries: + if order < 0: + raise ValueError("series order must be nonnegative") + product = _series_identity(order) + for stage in stages: + product = _series_multiply( + product, + _stage_exponential(stage, order), + ) + delta = _series_add( + product, + _series_scale(_series_identity(order), -1), + ) + logarithm: CubicWordSeries = [{} for _ in range(order + 1)] + power = _series_identity(order) + for exponent in range(1, order + 1): + power = _series_multiply(power, delta) + logarithm = _series_add( + logarithm, + _series_scale( + power, + Fraction(1 if exponent % 2 else -1, exponent), + ), + ) + return logarithm + + +def canonicalize_cubic_density( + registry: CoordinateRegistry, + operator: Mapping[SymplecticPauli, Cubic], +) -> CubicTerms: + result: CubicTerms = {} + for raw_pauli, coefficient in operator.items(): + pauli = canonicalize_symplectic_unit_cell(registry, raw_pauli) + _add_term(result, pauli, coefficient) + return result + + +def exact_log_e5_density( + stages: Sequence[CubicStage], +) -> tuple[CoordinateRegistry, CubicTerms]: + """Return the exact fifth-degree logarithm density.""" + + logarithm = cubic_formula_log_series(stages, 5) + evaluator = SymplecticDyadicLocalDensityEvaluator(shared_coordinates=True) + registry = evaluator.registries[0] + result: CubicTerms = {} + denominator = 5 * (1 << evaluator.denominator_exponent((0,) * 5)) + for word, word_coefficient in logarithm[5].items(): + for raw_pauli, numerator in evaluator.evaluate(word).items(): + pauli = canonicalize_symplectic_unit_cell(registry, raw_pauli) + _add_term( + result, + pauli, + word_coefficient * Fraction(numerator, denominator), + ) + return registry, result + + +def cubic_fragment_adjoint( + registry: CoordinateRegistry, + color: int, + operator: Mapping[SymplecticPauli, Cubic], +) -> CubicTerms: + """Apply one physical Heisenberg adjoint to an exact cubic map.""" + + candidate_bonds: set[CoordinateBond] = set() + for x_mask, z_mask in operator: + for site in _iter_set_bits(x_mask | z_mask): + coordinate = registry.coordinate(site) + candidate_bonds.add( + canonical_coordinate_bond( + coordinate, + matching_partner(coordinate, color), + ) + ) + + result: CubicTerms = {} + for bond in sorted(candidate_bonds): + first, second = (registry.site(coordinate) for coordinate in bond) + for bond_pauli in _symplectic_bond_terms(first, second): + for pauli, coefficient in operator.items(): + if not _symplectic_anticommutes(bond_pauli, pauli): + continue + phase, product_pauli = _symplectic_product_phase( + bond_pauli, pauli + ) + sign = 1 if phase == 1 else -1 + _add_term(result, product_pauli, sign * coefficient / 2) + return result + + +def exact_d5_density( + registry: CoordinateRegistry, + e5: Mapping[SymplecticPauli, Cubic], +) -> CubicTerms: + result: CubicTerms = {} + for color in range(4): + image = cubic_fragment_adjoint(registry, color, e5) + for pauli, coefficient in image.items(): + _add_term(result, pauli, 2 * coefficient) + return canonicalize_cubic_density(registry, result) + + +def exact_matching_density( + color: int, +) -> tuple[CoordinateRegistry, CubicTerms]: + """Return one exact two-by-two-cell matching Hamiltonian density.""" + + if color not in range(4): + raise ValueError("matching color must be 0, 1, 2 or 3") + evaluator = SymplecticDyadicLocalDensityEvaluator(shared_coordinates=True) + registry = evaluator.registries[0] + key = (color,) + denominator = 1 << evaluator.denominator_exponent(key) + return registry, canonicalize_cubic_density( + registry, + { + pauli: Cubic(Fraction(numerator, denominator), 0, 0) + for pauli, numerator in evaluator.evaluate(key).items() + }, + ) + + +def _add_cubic_operator( + target: CubicTerms, + source: Mapping[SymplecticPauli, Cubic], + scalar: Cubic | int | Fraction = 1, +) -> None: + for pauli, coefficient in source.items(): + _add_term(target, pauli, coefficient * scalar) + + +def conjugate_cubic_series_by_stage( + registry: CoordinateRegistry, + series: Sequence[Mapping[SymplecticPauli, Cubic]], + color: int, + coefficient: Cubic, +) -> list[CubicTerms]: + """Conjugate an exact local series by one matching exponential.""" + + order = len(series) - 1 + result: list[CubicTerms] = [{} for _ in range(order + 1)] + for degree, operator in enumerate(series): + power = dict(operator) + for nested_degree in range(order - degree + 1): + _add_cubic_operator( + result[degree + nested_degree], + power, + coefficient**nested_degree / factorial(nested_degree), + ) + if nested_degree < order - degree: + power = canonicalize_cubic_density( + registry, + cubic_fragment_adjoint(registry, color, power), + ) + return result + + +def exact_right_generator_stage_contribution( + stages: Sequence[CubicStage], + stage_index: int, + order: int, +) -> tuple[CoordinateRegistry, list[CubicTerms]]: + """Return one stage's exact contribution to ``i S' S^dagger``. + + The full right generator is the exact sum of this function over every + stage. Keeping stage contributions independent makes the expensive local + recurrence safe to distribute across Slurm array cells. + """ + + if order < 0: + raise ValueError("series order must be nonnegative") + if stage_index < 0 or stage_index >= len(stages): + raise IndexError("stage index is out of range") + stage = stages[stage_index] + registry, base = exact_matching_density(stage.fragment_index) + series: list[CubicTerms] = [{} for _ in range(order + 1)] + series[0] = { + pauli: stage.coefficient * coefficient + for pauli, coefficient in base.items() + } + for later in stages[stage_index + 1 :]: + series = conjugate_cubic_series_by_stage( + registry, + series, + later.fragment_index, + later.coefficient, + ) + return registry, [ + canonicalize_cubic_density(registry, degree) for degree in series + ] diff --git a/tracks/qcs/solutions/WangTheoPhys/issue128/src/trottercert/defect_series.py b/tracks/qcs/solutions/WangTheoPhys/issue128/src/trottercert/defect_series.py new file mode 100644 index 000000000..b226dcd93 --- /dev/null +++ b/tracks/qcs/solutions/WangTheoPhys/issue128/src/trottercert/defect_series.py @@ -0,0 +1,153 @@ +from __future__ import annotations + +from math import factorial +from typing import Sequence + +from .higher_order import ScalarStage +from .local_commutators import ( + CoordinateRegistry, + SymplecticDyadicLocalDensityEvaluator, + SymplecticPauli, + symplectic_local_fragment_adjoint, +) + + +FloatOperator = dict[SymplecticPauli, float] + + +def _pauli_coordinates( + registry: CoordinateRegistry, + pauli: SymplecticPauli, +) -> list[tuple[int, int, str]]: + x_mask, z_mask = pauli + sites = x_mask | z_mask + result: list[tuple[int, int, str]] = [] + while sites: + bit = sites & -sites + site = bit.bit_length() - 1 + x = bool(x_mask & bit) + z = bool(z_mask & bit) + op = "Y" if x and z else ("X" if x else "Z") + coordinate = registry.coordinate(site) + result.append((coordinate[0], coordinate[1], op)) + sites ^= bit + return result + + +def canonicalize_unit_cell_operator( + registry: CoordinateRegistry, + operator: FloatOperator, + unit_cell: tuple[int, int] = (2, 2), + *, + tolerance: float = 1e-18, +) -> FloatOperator: + result: FloatOperator = {} + step_x, step_y = unit_cell + for pauli, coefficient in operator.items(): + coordinates = _pauli_coordinates(registry, pauli) + if not coordinates: + key = (0, 0) + else: + min_x = min(x for x, _, _ in coordinates) + min_y = min(y for _, y, _ in coordinates) + shift_x = -(min_x - (min_x % step_x)) + shift_y = -(min_y - (min_y % step_y)) + x_mask = z_mask = 0 + for x, y, op in coordinates: + site = registry.site((x + shift_x, y + shift_y)) + bit = 1 << site + if op in {"X", "Y"}: + x_mask |= bit + if op in {"Z", "Y"}: + z_mask |= bit + key = (x_mask, z_mask) + updated = result.get(key, 0.0) + coefficient + if abs(updated) > tolerance: + result[key] = updated + else: + result.pop(key, None) + return result + + +def _add_scaled( + target: FloatOperator, + source: FloatOperator, + scalar: float, + *, + tolerance: float = 1e-18, +) -> None: + if not scalar: + return + for pauli, coefficient in source.items(): + updated = target.get(pauli, 0.0) + scalar * coefficient + if abs(updated) > tolerance: + target[pauli] = updated + else: + target.pop(pauli, None) + + +def normalized_local_adjoint( + registry: CoordinateRegistry, + color: int, + operator: FloatOperator, +) -> FloatOperator: + raw = symplectic_local_fragment_adjoint(registry, color, operator) + return canonicalize_unit_cell_operator( + registry, + {pauli: coefficient / 2 for pauli, coefficient in raw.items()}, + ) + + +def conjugate_series_by_stage( + registry: CoordinateRegistry, + series: Sequence[FloatOperator], + color: int, + coefficient: float, +) -> list[FloatOperator]: + order = len(series) - 1 + result: list[FloatOperator] = [{} for _ in range(order + 1)] + for degree, operator in enumerate(series): + power = operator + for nested_degree in range(order - degree + 1): + scalar = coefficient**nested_degree / factorial(nested_degree) + _add_scaled(result[degree + nested_degree], power, scalar) + if nested_degree < order - degree: + power = normalized_local_adjoint(registry, color, power) + return result + + +def right_generator_local_series( + stages_left_to_right: Sequence[ScalarStage], + order: int, +) -> list[FloatOperator]: + """Return coefficients of ``i S'(t) S(t)^dagger`` as cell densities.""" + + evaluator = SymplecticDyadicLocalDensityEvaluator(shared_coordinates=True) + registry = evaluator.registries[0] + base_operators = [] + for color in range(4): + base = { + pauli: numerator / 4 + for pauli, numerator in evaluator.evaluate((color,)).items() + } + base_operators.append( + canonicalize_unit_cell_operator(registry, base) + ) + + # Build S recursively by left multiplication: S_new = V S_old. + # Then i S_new' S_new^dagger = a H_color + Ad_V(G_old). + # This avoids replaying every prefix separately. + generator: list[FloatOperator] = [{} for _ in range(order + 1)] + for stage in stages_left_to_right: + generator = conjugate_series_by_stage( + registry, + generator, + stage.fragment_index, + float(stage.coefficient), + ) + _add_scaled( + generator[0], + base_operators[stage.fragment_index], + float(stage.coefficient), + ) + return generator diff --git a/tracks/qcs/solutions/WangTheoPhys/issue128/src/trottercert/formulas.py b/tracks/qcs/solutions/WangTheoPhys/issue128/src/trottercert/formulas.py new file mode 100644 index 000000000..e972ed105 --- /dev/null +++ b/tracks/qcs/solutions/WangTheoPhys/issue128/src/trottercert/formulas.py @@ -0,0 +1,94 @@ +from __future__ import annotations + +from dataclasses import dataclass +from fractions import Fraction +from typing import Sequence + +from .algebra import PauliSum, QComplex, commutator +from .series import exponential_series, logarithm_series, product_series + + +@dataclass(frozen=True) +class FormulaStage: + fragment: PauliSum + coefficient: Fraction + fragment_index: int + + +def strang_stages( + fragments: Sequence[PauliSum], + ordering: Sequence[int] | None = None, +) -> tuple[FormulaStage, ...]: + if len(fragments) < 2: + raise ValueError("Strang formula requires at least two fragments") + selected = tuple(ordering) if ordering is not None else tuple(range(len(fragments))) + if sorted(selected) != list(range(len(fragments))): + raise ValueError("ordering must be a permutation of fragment indices") + stages: list[FormulaStage] = [] + for index in selected[:-1]: + stages.append(FormulaStage(fragments[index], Fraction(1, 2), index)) + center = selected[-1] + stages.append(FormulaStage(fragments[center], Fraction(1), center)) + for index in reversed(selected[:-1]): + stages.append(FormulaStage(fragments[index], Fraction(1, 2), index)) + return tuple(stages) + + +def formula_log_series( + stages: Sequence[FormulaStage], + order: int = 3, +) -> list[PauliSum]: + exponentials = [ + exponential_series( + stage.fragment, + QComplex(0, -stage.coefficient), + order, + ) + for stage in stages + ] + return logarithm_series(product_series(exponentials, order)).coefficients + + +def formula_log_through_degree_three( + stages: Sequence[FormulaStage], +) -> tuple[PauliSum, PauliSum, PauliSum]: + """Compute log of the stage product by a graded BCH recurrence. + + This avoids expanding ordinary fragment products. If ``X`` is the + accumulated logarithm and ``Y`` is the next degree-one stage generator, + BCH through degree three is + + ``X + Y + [X,Y]/2 + ([X,[X,Y]] + [Y,[Y,X]])/12``. + """ + + degree_one = PauliSum.zero() + degree_two = PauliSum.zero() + degree_three = PauliSum.zero() + for stage in stages: + next_one = stage.fragment.scale(QComplex(0, -stage.coefficient)) + comm_one = commutator(degree_one, next_one) + new_three = ( + degree_three + + commutator(degree_two, next_one).scale(Fraction(1, 2)) + + commutator(degree_one, comm_one).scale(Fraction(1, 12)) + + commutator(next_one, commutator(next_one, degree_one)).scale( + Fraction(1, 12) + ) + ) + new_two = degree_two + comm_one.scale(Fraction(1, 2)) + degree_one = degree_one + next_one + degree_two = new_two + degree_three = new_three + return degree_one, degree_two, degree_three + + +def leading_effective_error(stages: Sequence[FormulaStage]) -> PauliSum: + """Return Hermitian E3 where log(S(t)) = -itH - it^3 E3 + O(t^5).""" + + _, degree_two, degree_three = formula_log_through_degree_three(stages) + if degree_two: + raise ArithmeticError("formula is not second order") + error = degree_three.scale(QComplex(0, 1)) + if not error.is_hermitian(): + raise ArithmeticError("leading effective error is not Hermitian") + return error diff --git a/tracks/qcs/solutions/WangTheoPhys/issue128/src/trottercert/hamiltonian.py b/tracks/qcs/solutions/WangTheoPhys/issue128/src/trottercert/hamiltonian.py new file mode 100644 index 000000000..a7dbd2314 --- /dev/null +++ b/tracks/qcs/solutions/WangTheoPhys/issue128/src/trottercert/hamiltonian.py @@ -0,0 +1,28 @@ +from __future__ import annotations + +from fractions import Fraction + +from .algebra import PauliString, PauliSum +from .lattice import SquareLattice + + +def heisenberg_bond(u: int, v: int) -> PauliSum: + result = PauliSum.zero() + for op in ("X", "Y", "Z"): + result += PauliSum.term(PauliString({u: op, v: op}), Fraction(1, 4)) + return result + + +def fragment_from_bonds(bonds: tuple[tuple[int, int], ...]) -> PauliSum: + result = PauliSum.zero() + for u, v in bonds: + result += heisenberg_bond(u, v) + return result + + +def four_matching_fragments(lattice: SquareLattice) -> tuple[PauliSum, ...]: + return tuple(fragment_from_bonds(group) for group in lattice.four_matchings()) + + +def full_heisenberg_hamiltonian(lattice: SquareLattice) -> PauliSum: + return fragment_from_bonds(lattice.bonds()) diff --git a/tracks/qcs/solutions/WangTheoPhys/issue128/src/trottercert/higher_order.py b/tracks/qcs/solutions/WangTheoPhys/issue128/src/trottercert/higher_order.py new file mode 100644 index 000000000..fba2322ce --- /dev/null +++ b/tracks/qcs/solutions/WangTheoPhys/issue128/src/trottercert/higher_order.py @@ -0,0 +1,516 @@ +from __future__ import annotations + +from dataclasses import dataclass +from functools import lru_cache +from itertools import product +from math import factorial +from typing import Iterable, Sequence + +import mpmath as mp +import numpy as np + +from .algebra import PauliSum, commutator +from .local_commutators import ( + LocalDensityEvaluator, + SymplecticDyadicLocalDensityEvaluator, +) + + +@dataclass(frozen=True) +class ScalarStage: + fragment_index: int + coefficient: mp.mpf + + +def _second_order_scalar_stages( + n_fragments: int, + scale: mp.mpf, +) -> list[ScalarStage]: + stages = [ + ScalarStage(index, scale / 2) + for index in range(n_fragments - 1) + ] + stages.append(ScalarStage(n_fragments - 1, scale)) + stages.extend( + ScalarStage(index, scale / 2) + for index in reversed(range(n_fragments - 1)) + ) + return stages + + +def merge_scalar_stages(stages: Sequence[ScalarStage]) -> tuple[ScalarStage, ...]: + merged: list[ScalarStage] = [] + for stage in stages: + if merged and merged[-1].fragment_index == stage.fragment_index: + previous = merged.pop() + coefficient = previous.coefficient + stage.coefficient + if coefficient: + merged.append(ScalarStage(stage.fragment_index, coefficient)) + else: + merged.append(stage) + return tuple(merged) + + +def fourth_order_suzuki_stages(n_fragments: int) -> tuple[ScalarStage, ...]: + """Five-copy Suzuki fractal S4 built from the symmetric S2 kernel.""" + + mp.mp.dps = max(mp.mp.dps, 80) + u = 1 / (4 - mp.root(4, 3)) + scales = (u, u, 1 - 4 * u, u, u) + stages: list[ScalarStage] = [] + for scale in scales: + stages.extend(_second_order_scalar_stages(n_fragments, scale)) + return merge_scalar_stages(stages) + + +def fourth_order_triple_jump_stages(n_fragments: int) -> tuple[ScalarStage, ...]: + """Three-copy fourth-order symmetric composition of S2.""" + + mp.mp.dps = max(mp.mp.dps, 80) + a = 1 / (2 - mp.root(2, 3)) + b = 1 - 2 * a + stages: list[ScalarStage] = [] + for scale in (a, b, a): + stages.extend(_second_order_scalar_stages(n_fragments, scale)) + return merge_scalar_stages(stages) + + +def weak_compositions(total: int, length: int) -> Iterable[tuple[int, ...]]: + if length == 1: + yield (total,) + return + for first in range(total + 1): + for tail in weak_compositions(total - first, length - 1): + yield (first,) + tail + + +def _multinomial(total: int, composition: Sequence[int]) -> int: + result = factorial(total) + for value in composition: + result //= factorial(value) + return result + + +def higher_order_triangle_weights( + stages_left_to_right: Sequence[ScalarStage], + order: int, + center: int | None = None, +) -> dict[tuple[int, ...], mp.mpf]: + """Aggregate Schubert--Mendl theorem weights by base-fragment sequence. + + The returned key is ``(outermost, ..., innermost, base)``. The norm of + the corresponding pure nested commutator is multiplied by the positive + weight. Expanding ``B_j`` by a triangle inequality keeps this routine + conservative and easy to verify. + """ + + # Their A_1 is the rightmost exponential. + stages = tuple(reversed(stages_left_to_right)) + count = len(stages) + s = center if center is not None else (count + 1) // 2 + if not 1 <= s <= count: + raise ValueError("center must use one-based stage indexing") + weights: dict[tuple[int, ...], mp.mpf] = {} + + def add_range(j: int, indices: tuple[int, ...], composition: tuple[int, ...]) -> None: + outer: list[int] = [] + coefficient = mp.mpf(_multinomial(order, composition)) + for stage_index, power in zip(indices, composition): + coefficient *= abs(stages[stage_index - 1].coefficient) ** power + # ad operators appear in the theorem in the same order as indices. + for stage_index, power in zip(indices, composition): + outer.extend([stages[stage_index - 1].fragment_index] * power) + for base_index in range(1, j): + base_stage = stages[base_index - 1] + key = tuple(outer) + (base_stage.fragment_index,) + weights[key] = weights.get(key, mp.mpf("0")) + ( + coefficient * abs(base_stage.coefficient) + ) + + for j in range(2, s + 1): + indices = tuple(range(s, j - 1, -1)) + # q_j is the final entry because indices descend s,...,j. + for composition in weak_compositions(order, len(indices)): + if composition[-1] == 0: + continue + add_range(j, indices, composition) + + for j in range(s + 1, count + 1): + indices = tuple(range(s + 1, j + 1)) + # q_j is the final entry because indices ascend s+1,...,j. + for composition in weak_compositions(order, len(indices)): + if composition[-1] == 0: + continue + add_range(j, indices, composition) + return weights + + +def nested_commutator( + fragments: Sequence[PauliSum], + key: tuple[int, ...], + cache: dict[tuple[int, ...], PauliSum] | None = None, +) -> PauliSum: + """Evaluate a key ``(outermost,...,innermost,base)`` exactly.""" + + memo = cache if cache is not None else {} + if key in memo: + return memo[key] + if len(key) == 1: + result = fragments[key[0]] + else: + result = commutator( + fragments[key[0]], + nested_commutator(fragments, key[1:], memo), + ) + memo[key] = result + return result + + +def fourth_order_pauli_l1_constant( + fragments: Sequence[PauliSum], +) -> tuple[mp.mpf, dict[tuple[int, ...], mp.mpf]]: + stages = fourth_order_suzuki_stages(len(fragments)) + weights = higher_order_triangle_weights(stages, order=4) + cache: dict[tuple[int, ...], PauliSum] = {} + total = mp.mpf("0") + for key, weight in weights.items(): + operator = nested_commutator(fragments, key, cache) + total += weight * operator.pauli_l1() + return total / factorial(5), weights + + +def fourth_order_local_pauli_l1_density( + n_fragments: int = 4, +) -> tuple[mp.mpf, dict[tuple[int, ...], mp.mpf]]: + """Uniform 2x2-cell fourth-order bound for the four matching model.""" + + if n_fragments != 4: + raise ValueError("local matching density currently supports four fragments") + stages = fourth_order_suzuki_stages(n_fragments) + weights = higher_order_triangle_weights(stages, order=4) + evaluator = LocalDensityEvaluator() + total = mp.mpf("0") + for key, weight in weights.items(): + if key[-2] == key[-1]: + continue + norm = evaluator.pauli_l1_density(key) + total += weight * int(norm.numerator) / int( + norm.denominator + ) + return total / factorial(5), weights + + +def local_pauli_l1_constant_from_norms( + stages: Sequence[ScalarStage], + norm_density: dict[tuple[int, ...], Fraction], + *, + order: int = 4, +) -> mp.mpf: + weights = higher_order_triangle_weights(stages, order) + total = mp.mpf("0") + for key, weight in weights.items(): + norm = norm_density.get(key, Fraction()) + total += weight * int(norm.numerator) / int(norm.denominator) + return total / factorial(order + 1) + + +def higher_order_combined_local_l1_density( + stages_left_to_right: Sequence[ScalarStage], + *, + order: int = 4, + center: int | None = None, +) -> mp.mpf: + """Evaluate the theorem while retaining cancellations inside each B_j.""" + + stages = tuple(reversed(stages_left_to_right)) + count = len(stages) + s = center if center is not None else (count + 1) // 2 + evaluator = LocalDensityEvaluator() + total = mp.mpf("0") + + def evaluate_term( + j: int, + indices: tuple[int, ...], + composition: tuple[int, ...], + ) -> mp.mpf: + outer: list[int] = [] + scalar = mp.mpf(_multinomial(order, composition)) + for stage_index, power in zip(indices, composition): + stage = stages[stage_index - 1] + scalar *= stage.coefficient**power + outer.extend([stage.fragment_index] * power) + base_coefficients = [mp.mpf("0") for _ in range(4)] + for base_index in range(1, j): + base = stages[base_index - 1] + base_coefficients[base.fragment_index] += base.coefficient + pauli_coefficients: dict[object, mp.mpf] = {} + for fragment_index, base_coefficient in enumerate(base_coefficients): + if not base_coefficient: + continue + operator = evaluator.evaluate(tuple(outer) + (fragment_index,)) + for pauli, coefficient in operator.terms.items(): + if coefficient.imag: + raise ArithmeticError("degree-five commutator should be Hermitian") + pauli_coefficients[pauli] = pauli_coefficients.get( + pauli, mp.mpf("0") + ) + base_coefficient * mp.mpf(coefficient.real.numerator) / mp.mpf( + coefficient.real.denominator + ) + return abs(scalar) * sum(abs(value) for value in pauli_coefficients.values()) + + for j in range(2, s + 1): + indices = tuple(range(s, j - 1, -1)) + for composition in weak_compositions(order, len(indices)): + if composition[-1]: + total += evaluate_term(j, indices, composition) + for j in range(s + 1, count + 1): + indices = tuple(range(s + 1, j + 1)) + for composition in weak_compositions(order, len(indices)): + if composition[-1]: + total += evaluate_term(j, indices, composition) + return total / factorial(order + 1) + + +def higher_order_combined_local_l1_density_fast( + stages_left_to_right: Sequence[ScalarStage], + *, + order: int = 4, + center: int | None = None, +) -> float: + """Vectorized discovery evaluation retaining cancellations inside B_j.""" + + stages = tuple(reversed(stages_left_to_right)) + count = len(stages) + s = center if center is not None else (count + 1) // 2 + grouped_terms: dict[ + tuple[int, ...], list[tuple[float, tuple[float, ...]]] + ] = {} + + def collect( + j: int, + indices: tuple[int, ...], + composition: tuple[int, ...], + ) -> None: + outer: list[int] = [] + scalar = float(_multinomial(order, composition)) + for stage_index, power in zip(indices, composition): + stage = stages[stage_index - 1] + scalar *= float(stage.coefficient) ** power + outer.extend([stage.fragment_index] * power) + base = [0.0] * 4 + for base_index in range(1, j): + stage = stages[base_index - 1] + base[stage.fragment_index] += float(stage.coefficient) + grouped_terms.setdefault(tuple(outer), []).append( + (abs(scalar), tuple(base)) + ) + + for j in range(2, s + 1): + indices = tuple(range(s, j - 1, -1)) + for composition in weak_compositions(order, len(indices)): + if composition[-1]: + collect(j, indices, composition) + for j in range(s + 1, count + 1): + indices = tuple(range(s + 1, j + 1)) + for composition in weak_compositions(order, len(indices)): + if composition[-1]: + collect(j, indices, composition) + + evaluator = LocalDensityEvaluator() + total = 0.0 + for outer, terms in grouped_terms.items(): + operators = [evaluator.evaluate(outer + (base,)) for base in range(4)] + paulis = sorted(set().union(*(set(operator.terms) for operator in operators))) + matrix = np.zeros((len(paulis), 4), dtype=np.float64) + for column, operator in enumerate(operators): + for row, pauli in enumerate(paulis): + coefficient = operator.terms.get(pauli) + if coefficient is not None: + matrix[row, column] = float(coefficient.real) + scalars = np.asarray([term[0] for term in terms], dtype=np.float64) + base_matrix = np.asarray([term[1] for term in terms], dtype=np.float64).T + norms = np.abs(matrix @ base_matrix).sum(axis=0) + total += float(scalars @ norms) + return total / factorial(order + 1) + + +def higher_order_combined_local_l1_density_symplectic( + stages_left_to_right: Sequence[ScalarStage], + *, + order: int = 4, + center: int | None = None, + evaluator: SymplecticDyadicLocalDensityEvaluator | None = None, +) -> float: + """Fast discovery bound retaining exact Pauli cancellation inside ``B_j``. + + Local nested commutators are evaluated exactly as integer coefficient + vectors. Formula coefficients are converted to double precision only + at the final small linear combinations, so this is a discovery value, + not yet an outward-rounded certificate. + """ + + stages = tuple(reversed(stages_left_to_right)) + count = len(stages) + s = center if center is not None else (count + 1) // 2 + grouped_terms: dict[ + tuple[int, ...], list[tuple[float, tuple[float, ...]]] + ] = {} + + def collect( + j: int, + indices: tuple[int, ...], + composition: tuple[int, ...], + ) -> None: + outer: list[int] = [] + scalar = float(_multinomial(order, composition)) + for stage_index, power in zip(indices, composition): + stage = stages[stage_index - 1] + scalar *= float(stage.coefficient) ** power + outer.extend([stage.fragment_index] * power) + base = [0.0] * 4 + for base_index in range(1, j): + stage = stages[base_index - 1] + base[stage.fragment_index] += float(stage.coefficient) + grouped_terms.setdefault(tuple(outer), []).append( + (abs(scalar), tuple(base)) + ) + + for j in range(2, s + 1): + indices = tuple(range(s, j - 1, -1)) + for composition in weak_compositions(order, len(indices)): + if composition[-1]: + collect(j, indices, composition) + for j in range(s + 1, count + 1): + indices = tuple(range(s + 1, j + 1)) + for composition in weak_compositions(order, len(indices)): + if composition[-1]: + collect(j, indices, composition) + + # Combining different base fragments requires one coordinate-to-bit + # registry; otherwise equal physical Pauli strings can receive unrelated + # bit labels and their cancellations would be destroyed. + local = evaluator or SymplecticDyadicLocalDensityEvaluator( + shared_coordinates=True + ) + denominator = float(1 << (order + 2)) + total = 0.0 + for outer, terms in grouped_terms.items(): + operators = [local.evaluate(outer + (base,)) for base in range(4)] + paulis = set().union(*(operator.keys() for operator in operators)) + row_for_pauli = {pauli: row for row, pauli in enumerate(paulis)} + matrix = np.zeros((len(paulis), 4), dtype=np.int32) + for column, operator in enumerate(operators): + for pauli, coefficient in operator.items(): + matrix[row_for_pauli[pauli], column] = coefficient + scalars = np.asarray([term[0] for term in terms], dtype=np.float64) + base_matrix = np.asarray([term[1] for term in terms], dtype=np.float64).T + norms = np.abs(matrix @ base_matrix).sum(axis=0) / denominator + total += float(scalars @ norms) + return total / factorial(order + 1) + + +def _pairwise_anticommuting_float_bound( + paulis: Sequence[tuple[int, int]], + coefficients: np.ndarray, +) -> float: + """Greedily pair anticommuting Paulis and apply Euclidean pair norms.""" + + order = np.argsort(-np.abs(coefficients), kind="stable") + used = np.zeros(len(paulis), dtype=np.bool_) + total = 0.0 + for position, index_value in enumerate(order): + index = int(index_value) + if used[index] or coefficients[index] == 0: + continue + used[index] = True + left_x, left_z = paulis[index] + partner: int | None = None + for candidate_value in order[position + 1 :]: + candidate = int(candidate_value) + if used[candidate] or coefficients[candidate] == 0: + continue + right_x, right_z = paulis[candidate] + parity = ( + (left_x & right_z).bit_count() + + (left_z & right_x).bit_count() + ) & 1 + if parity: + partner = candidate + break + if partner is None: + total += abs(float(coefficients[index])) + else: + used[partner] = True + total += float( + np.hypot(coefficients[index], coefficients[partner]) + ) + return total + + +def higher_order_combined_local_pair_density_symplectic( + stages_left_to_right: Sequence[ScalarStage], + *, + order: int = 4, + center: int | None = None, + evaluator: SymplecticDyadicLocalDensityEvaluator | None = None, +) -> float: + """Discovery bound with ``B_j`` cancellation and anticommuting pairs.""" + + stages = tuple(reversed(stages_left_to_right)) + count = len(stages) + s = center if center is not None else (count + 1) // 2 + grouped: dict[ + tuple[int, ...], + dict[tuple[float, ...], float], + ] = {} + + def collect( + j: int, + indices: tuple[int, ...], + composition: tuple[int, ...], + ) -> None: + outer: list[int] = [] + scalar = float(_multinomial(order, composition)) + for stage_index, power in zip(indices, composition): + stage = stages[stage_index - 1] + scalar *= float(stage.coefficient) ** power + outer.extend([stage.fragment_index] * power) + base = [0.0] * 4 + for base_index in range(1, j): + stage = stages[base_index - 1] + base[stage.fragment_index] += float(stage.coefficient) + base_key = tuple(base) + by_base = grouped.setdefault(tuple(outer), {}) + by_base[base_key] = by_base.get(base_key, 0.0) + abs(scalar) + + for j in range(2, s + 1): + indices = tuple(range(s, j - 1, -1)) + for composition in weak_compositions(order, len(indices)): + if composition[-1]: + collect(j, indices, composition) + for j in range(s + 1, count + 1): + indices = tuple(range(s + 1, j + 1)) + for composition in weak_compositions(order, len(indices)): + if composition[-1]: + collect(j, indices, composition) + + local = evaluator or SymplecticDyadicLocalDensityEvaluator( + shared_coordinates=True + ) + denominator = float(1 << (order + 2)) + total = 0.0 + for outer, by_base in grouped.items(): + operators = [local.evaluate(outer + (base,)) for base in range(4)] + paulis = tuple(set().union(*(operator.keys() for operator in operators))) + row_for_pauli = {pauli: row for row, pauli in enumerate(paulis)} + matrix = np.zeros((len(paulis), 4), dtype=np.int32) + for column, operator in enumerate(operators): + for pauli, coefficient in operator.items(): + matrix[row_for_pauli[pauli], column] = coefficient + for base, scalar_weight in by_base.items(): + coefficients = matrix @ np.asarray(base, dtype=np.float64) + total += scalar_weight * _pairwise_anticommuting_float_bound( + paulis, + coefficients, + ) / denominator + return total / factorial(order + 1) diff --git a/tracks/qcs/solutions/WangTheoPhys/issue128/src/trottercert/hpc_artifacts.py b/tracks/qcs/solutions/WangTheoPhys/issue128/src/trottercert/hpc_artifacts.py new file mode 100644 index 000000000..169b6674e --- /dev/null +++ b/tracks/qcs/solutions/WangTheoPhys/issue128/src/trottercert/hpc_artifacts.py @@ -0,0 +1,212 @@ +from __future__ import annotations + +from collections.abc import Mapping, Sequence +from fractions import Fraction +import gzip +import hashlib +import json +import os +from pathlib import Path +import tempfile +from typing import Any + +from .cubic_field import Cubic +from .cubic_local import CubicTerms +from .local_commutators import CoordinateRegistry, SymplecticPauli, _iter_set_bits + + +CoordinatePauli = tuple[tuple[int, int, str], ...] +EncodedCubicTerm = list[object] +CoordinateCubicTerms = dict[CoordinatePauli, Cubic] + + +def _pauli_coordinates( + registry: CoordinateRegistry, + pauli: SymplecticPauli, +) -> CoordinatePauli: + x_mask, z_mask = pauli + result: list[tuple[int, int, str]] = [] + for site in _iter_set_bits(x_mask | z_mask): + bit = 1 << site + x = bool(x_mask & bit) + z = bool(z_mask & bit) + operator = "Y" if x and z else ("X" if x else "Z") + coordinate = registry.coordinate(site) + result.append((coordinate[0], coordinate[1], operator)) + return tuple(sorted(result)) + + +def _pair(value: Fraction) -> list[int]: + return [value.numerator, value.denominator] + + +def _encode_cubic(value: Cubic) -> list[list[int]]: + return [_pair(value.a0), _pair(value.a1), _pair(value.a2)] + + +def _decode_fraction(value: object) -> Fraction: + if ( + not isinstance(value, list) + or len(value) != 2 + or isinstance(value[0], bool) + or isinstance(value[1], bool) + or not isinstance(value[0], int) + or not isinstance(value[1], int) + or value[1] <= 0 + ): + raise ValueError("malformed rational pair") + return Fraction(value[0], value[1]) + + +def _decode_cubic(value: object) -> Cubic: + if not isinstance(value, list) or len(value) != 3: + raise ValueError("malformed cubic coefficient") + return Cubic(*(_decode_fraction(part) for part in value)) + + +def cubic_to_json(value: Cubic) -> list[list[int]]: + return _encode_cubic(value) + + +def cubic_from_json(value: object) -> Cubic: + return _decode_cubic(value) + + +def coordinate_encode_terms( + registry: CoordinateRegistry, + terms: Mapping[SymplecticPauli, Cubic], +) -> list[EncodedCubicTerm]: + encoded = [ + [ + [[x, y, operator] for x, y, operator in _pauli_coordinates(registry, pauli)], + _encode_cubic(coefficient), + ] + for pauli, coefficient in terms.items() + ] + return sorted(encoded, key=lambda term: term[0]) + + +def coordinate_decode_terms( + terms: object, +) -> CoordinateCubicTerms: + if not isinstance(terms, list): + raise ValueError("encoded terms must be a list") + result: CoordinateCubicTerms = {} + for term in terms: + if not isinstance(term, list) or len(term) != 2: + raise ValueError("malformed encoded term") + raw_key, raw_coefficient = term + if not isinstance(raw_key, list): + raise ValueError("malformed coordinate Pauli key") + key_parts: list[tuple[int, int, str]] = [] + for part in raw_key: + if ( + not isinstance(part, list) + or len(part) != 3 + or isinstance(part[0], bool) + or isinstance(part[1], bool) + or not isinstance(part[0], int) + or not isinstance(part[1], int) + or part[2] not in {"X", "Y", "Z"} + ): + raise ValueError("malformed coordinate Pauli factor") + key_parts.append((part[0], part[1], part[2])) + key = tuple(sorted(key_parts)) + if key in result: + raise ValueError("duplicate coordinate Pauli term") + coefficient = _decode_cubic(raw_coefficient) + if coefficient == Cubic.zero(): + raise ValueError("encoded shard contains a zero coefficient") + result[key] = coefficient + return result + + +def coordinate_terms_to_json( + terms: Mapping[CoordinatePauli, Cubic], +) -> list[EncodedCubicTerm]: + return [ + [ + [[x, y, operator] for x, y, operator in key], + _encode_cubic(terms[key]), + ] + for key in sorted(terms) + ] + + +def coordinate_encode_series( + registry: CoordinateRegistry, + series: Sequence[Mapping[SymplecticPauli, Cubic]], +) -> list[list[EncodedCubicTerm]]: + return [coordinate_encode_terms(registry, degree) for degree in series] + + +def merge_coordinate_series( + shards: Sequence[Sequence[object]], +) -> list[CoordinateCubicTerms]: + if not shards: + return [] + order = len(shards[0]) + if any(len(shard) != order for shard in shards): + raise ValueError("coordinate series orders differ") + result: list[CoordinateCubicTerms] = [{} for _ in range(order)] + for shard in shards: + for degree, raw_terms in enumerate(shard): + for key, coefficient in coordinate_decode_terms(raw_terms).items(): + updated = result[degree].get(key, Cubic.zero()) + coefficient + if updated == Cubic.zero(): + result[degree].pop(key, None) + else: + result[degree][key] = updated + return result + + +def _canonical_json_bytes(payload: object) -> bytes: + return (json.dumps(payload, sort_keys=True, separators=(",", ":")) + "\n").encode() + + +def _atomic_write(path: Path, raw: bytes) -> None: + path.parent.mkdir(parents=True, exist_ok=True) + descriptor, temporary_name = tempfile.mkstemp( + prefix=f".{path.name}.", suffix=".tmp", dir=path.parent + ) + temporary = Path(temporary_name) + try: + with os.fdopen(descriptor, "wb") as handle: + handle.write(raw) + handle.flush() + os.fsync(handle.fileno()) + temporary.replace(path) + except BaseException: + temporary.unlink(missing_ok=True) + raise + + +def write_shard_gzip(path: str | Path, payload: object) -> None: + raw = gzip.compress(_canonical_json_bytes(payload), compresslevel=9, mtime=0) + _atomic_write(Path(path), raw) + + +def read_shard_gzip(path: str | Path) -> dict[str, Any]: + raw = Path(path).read_bytes() + try: + payload = json.loads(gzip.decompress(raw)) + except (OSError, json.JSONDecodeError, UnicodeDecodeError) as error: + raise ValueError("shard is not valid canonical gzip JSON") from error + if not isinstance(payload, dict): + raise ValueError("shard root must be an object") + expected = gzip.compress(_canonical_json_bytes(payload), compresslevel=9, mtime=0) + if expected != raw: + raise ValueError("shard gzip JSON is not canonical") + return payload + + +def write_manifest_atomic(path: str | Path, payload: object) -> None: + _atomic_write(Path(path), _canonical_json_bytes(payload)) + + +def sha256_file(path: str | Path) -> str: + digest = hashlib.sha256() + with Path(path).open("rb") as handle: + for block in iter(lambda: handle.read(1 << 20), b""): + digest.update(block) + return digest.hexdigest() diff --git a/tracks/qcs/solutions/WangTheoPhys/issue128/src/trottercert/intervals.py b/tracks/qcs/solutions/WangTheoPhys/issue128/src/trottercert/intervals.py new file mode 100644 index 000000000..40000abd2 --- /dev/null +++ b/tracks/qcs/solutions/WangTheoPhys/issue128/src/trottercert/intervals.py @@ -0,0 +1,146 @@ +from __future__ import annotations + +from dataclasses import dataclass +from fractions import Fraction +from math import floor + + +@dataclass(frozen=True, slots=True) +class RationalInterval: + lower: Fraction + upper: Fraction + + def __post_init__(self) -> None: + if self.lower > self.upper: + raise ValueError("interval lower endpoint exceeds upper endpoint") + + @classmethod + def point(cls, value: int | Fraction) -> RationalInterval: + exact = value if isinstance(value, Fraction) else Fraction(value) + return cls(exact, exact) + + def __add__( + self, + other: RationalInterval | int | Fraction, + ) -> RationalInterval: + rhs = other if isinstance(other, RationalInterval) else self.point(other) + return RationalInterval(self.lower + rhs.lower, self.upper + rhs.upper) + + __radd__ = __add__ + + def __neg__(self) -> RationalInterval: + return RationalInterval(-self.upper, -self.lower) + + def __sub__( + self, + other: RationalInterval | int | Fraction, + ) -> RationalInterval: + rhs = other if isinstance(other, RationalInterval) else self.point(other) + return self + (-rhs) + + def __rsub__( + self, + other: RationalInterval | int | Fraction, + ) -> RationalInterval: + return self.point(other) - self + + def __mul__( + self, + other: RationalInterval | int | Fraction, + ) -> RationalInterval: + rhs = other if isinstance(other, RationalInterval) else self.point(other) + values = ( + self.lower * rhs.lower, + self.lower * rhs.upper, + self.upper * rhs.lower, + self.upper * rhs.upper, + ) + return RationalInterval(min(values), max(values)) + + __rmul__ = __mul__ + + def reciprocal(self) -> RationalInterval: + if self.lower <= 0 <= self.upper: + raise ZeroDivisionError("interval contains zero") + return RationalInterval(1 / self.upper, 1 / self.lower) + + def __truediv__( + self, + other: RationalInterval | int | Fraction, + ) -> RationalInterval: + rhs = other if isinstance(other, RationalInterval) else self.point(other) + return self * rhs.reciprocal() + + def __pow__(self, exponent: int) -> RationalInterval: + if exponent < 0: + return (self.reciprocal()) ** (-exponent) + result = self.point(1) + base = self + power = exponent + while power: + if power & 1: + result = result * base + base = base * base + power >>= 1 + return result + + def abs_upper(self) -> Fraction: + return max(abs(self.lower), abs(self.upper)) + + def abs_lower(self) -> Fraction: + if self.lower <= 0 <= self.upper: + return Fraction() + return min(abs(self.lower), abs(self.upper)) + + def midpoint(self) -> Fraction: + return (self.lower + self.upper) / 2 + + +def cube_root_four_interval(decimal_digits: int = 24) -> RationalInterval: + """Return a rational enclosure of the real cube root of four.""" + + return nth_root_four_interval(3, decimal_digits) + + +def nth_root_four_interval( + root_degree: int, + decimal_digits: int = 24, +) -> RationalInterval: + """Return a rational enclosure of the positive ``root_degree`` root of four.""" + + if root_degree < 1: + raise ValueError("root degree must be positive") + if decimal_digits < 1: + raise ValueError("decimal_digits must be positive") + denominator = 10**decimal_digits + low = denominator + high = 4 * denominator + target = 4 * denominator**root_degree + while low + 1 < high: + middle = (low + high) // 2 + if middle**root_degree <= target: + low = middle + else: + high = middle + return RationalInterval( + Fraction(low, denominator), + Fraction(high, denominator), + ) + + +def outward_quantize( + interval: RationalInterval, + denominator: int, +) -> RationalInterval: + """Round an interval outward onto a fixed rational grid.""" + + if denominator <= 0: + raise ValueError("denominator must be positive") + lower_scaled = interval.lower * denominator + upper_scaled = interval.upper * denominator + lower_integer = lower_scaled.numerator // lower_scaled.denominator + upper_integer = -((-upper_scaled.numerator) // upper_scaled.denominator) + return RationalInterval( + Fraction(lower_integer, denominator), + Fraction(upper_integer, denominator), + ) diff --git a/tracks/qcs/solutions/WangTheoPhys/issue128/src/trottercert/lattice.py b/tracks/qcs/solutions/WangTheoPhys/issue128/src/trottercert/lattice.py new file mode 100644 index 000000000..612ec501c --- /dev/null +++ b/tracks/qcs/solutions/WangTheoPhys/issue128/src/trottercert/lattice.py @@ -0,0 +1,55 @@ +from __future__ import annotations + +from dataclasses import dataclass + + +@dataclass(frozen=True, slots=True) +class SquareLattice: + length: int + + def __post_init__(self) -> None: + if self.length < 4 or self.length % 2: + raise ValueError("periodic matching lattice requires even L >= 4") + + @property + def n_sites(self) -> int: + return self.length * self.length + + def site(self, x: int, y: int) -> int: + return (y % self.length) * self.length + (x % self.length) + + def coordinates(self, site: int) -> tuple[int, int]: + if not 0 <= site < self.n_sites: + raise ValueError("site outside lattice") + return site % self.length, site // self.length + + @staticmethod + def canonical_bond(u: int, v: int) -> tuple[int, int]: + if u == v: + raise ValueError("self bonds are not allowed") + return (u, v) if u < v else (v, u) + + def bonds(self) -> tuple[tuple[int, int], ...]: + edges: set[tuple[int, int]] = set() + for y in range(self.length): + for x in range(self.length): + u = self.site(x, y) + edges.add(self.canonical_bond(u, self.site(x + 1, y))) + edges.add(self.canonical_bond(u, self.site(x, y + 1))) + return tuple(sorted(edges)) + + def four_matchings(self) -> tuple[tuple[tuple[int, int], ...], ...]: + groups: list[list[tuple[int, int]]] = [[], [], [], []] + for y in range(self.length): + for x in range(0, self.length, 2): + groups[0].append(self.canonical_bond(self.site(x, y), self.site(x + 1, y))) + groups[1].append( + self.canonical_bond(self.site(x + 1, y), self.site(x + 2, y)) + ) + for x in range(self.length): + for y in range(0, self.length, 2): + groups[2].append(self.canonical_bond(self.site(x, y), self.site(x, y + 1))) + groups[3].append( + self.canonical_bond(self.site(x, y + 1), self.site(x, y + 2)) + ) + return tuple(tuple(sorted(set(group))) for group in groups) diff --git a/tracks/qcs/solutions/WangTheoPhys/issue128/src/trottercert/lie_series.py b/tracks/qcs/solutions/WangTheoPhys/issue128/src/trottercert/lie_series.py new file mode 100644 index 000000000..06814e0ef --- /dev/null +++ b/tracks/qcs/solutions/WangTheoPhys/issue128/src/trottercert/lie_series.py @@ -0,0 +1,124 @@ +from __future__ import annotations + +from math import factorial +from typing import Sequence + +import mpmath as mp + +from .higher_order import ScalarStage +from .local_commutators import SymplecticDyadicLocalDensityEvaluator + + +Word = tuple[int, ...] +WordPolynomial = dict[Word, mp.mpf] +WordSeries = list[WordPolynomial] + + +def _series_identity(order: int) -> WordSeries: + result: WordSeries = [{} for _ in range(order + 1)] + result[0][()] = mp.mpf(1) + return result + + +def _series_add(left: WordSeries, right: WordSeries) -> WordSeries: + result: WordSeries = [] + for left_degree, right_degree in zip(left, right): + terms = dict(left_degree) + for word, coefficient in right_degree.items(): + updated = terms.get(word, mp.mpf(0)) + coefficient + if updated: + terms[word] = updated + else: + terms.pop(word, None) + result.append(terms) + return result + + +def _series_scale(series: WordSeries, scalar: mp.mpf) -> WordSeries: + return [ + {word: scalar * coefficient for word, coefficient in degree.items()} + for degree in series + ] + + +def _series_multiply(left: WordSeries, right: WordSeries) -> WordSeries: + order = len(left) - 1 + result: WordSeries = [{} for _ in range(order + 1)] + for degree in range(order + 1): + terms = result[degree] + for left_degree in range(degree + 1): + for left_word, left_coefficient in left[left_degree].items(): + for right_word, right_coefficient in right[degree - left_degree].items(): + word = left_word + right_word + terms[word] = terms.get(word, mp.mpf(0)) + ( + left_coefficient * right_coefficient + ) + return result + + +def abstract_exponential_series( + fragment_index: int, + coefficient: mp.mpf, + order: int, +) -> WordSeries: + result: WordSeries = [{} for _ in range(order + 1)] + for degree in range(order + 1): + result[degree][(fragment_index,) * degree] = ( + coefficient**degree / factorial(degree) + ) + return result + + +def abstract_formula_log_series( + stages: Sequence[ScalarStage], + order: int, +) -> WordSeries: + product = _series_identity(order) + for stage in stages: + product = _series_multiply( + product, + abstract_exponential_series( + stage.fragment_index, + stage.coefficient, + order, + ), + ) + delta = _series_add(product, _series_scale(_series_identity(order), mp.mpf(-1))) + logarithm: WordSeries = [{} for _ in range(order + 1)] + power = _series_identity(order) + for exponent in range(1, order + 1): + power = _series_multiply(power, delta) + logarithm = _series_add( + logarithm, + _series_scale(power, mp.mpf(1 if exponent % 2 else -1) / exponent), + ) + return logarithm + + +def leading_lie_local_coefficients( + stages: Sequence[ScalarStage], + degree: int, +) -> dict[tuple[int, int], mp.mpf]: + """Map a homogeneous log coefficient to its local Pauli density. + + The Dynkin--Specht--Wever projection maps every word to its right-nested + commutator and divides by the homogeneous degree. + """ + + logarithm = abstract_formula_log_series(stages, degree) + evaluator = SymplecticDyadicLocalDensityEvaluator(shared_coordinates=True) + scale = mp.mpf(1) / (degree * (1 << (degree + 1))) + coefficients: dict[tuple[int, int], mp.mpf] = {} + for word, word_coefficient in logarithm[degree].items(): + if not word_coefficient: + continue + operator = evaluator.evaluate(word) + for pauli, numerator in operator.items(): + updated = coefficients.get(pauli, mp.mpf(0)) + ( + word_coefficient * numerator * scale + ) + if updated: + coefficients[pauli] = updated + else: + coefficients.pop(pauli, None) + return coefficients diff --git a/tracks/qcs/solutions/WangTheoPhys/issue128/src/trottercert/local_commutators.py b/tracks/qcs/solutions/WangTheoPhys/issue128/src/trottercert/local_commutators.py new file mode 100644 index 000000000..feed35e8a --- /dev/null +++ b/tracks/qcs/solutions/WangTheoPhys/issue128/src/trottercert/local_commutators.py @@ -0,0 +1,397 @@ +from __future__ import annotations + +from dataclasses import dataclass, field +from fractions import Fraction +from typing import Iterable, Sequence + +from .algebra import ( + PauliString, + PauliSum, + commutator, + pauli_strings_commute, +) + + +Coordinate = tuple[int, int] +CoordinateBond = tuple[Coordinate, Coordinate] + + +@dataclass +class CoordinateRegistry: + coordinate_to_site: dict[Coordinate, int] = field(default_factory=dict) + site_to_coordinate: list[Coordinate] = field(default_factory=list) + + def site(self, coordinate: Coordinate) -> int: + if coordinate not in self.coordinate_to_site: + self.coordinate_to_site[coordinate] = len(self.site_to_coordinate) + self.site_to_coordinate.append(coordinate) + return self.coordinate_to_site[coordinate] + + def coordinate(self, site: int) -> Coordinate: + return self.site_to_coordinate[site] + + +def matching_partner(coordinate: Coordinate, color: int) -> Coordinate: + x, y = coordinate + if color == 0: + return (x + 1, y) if x % 2 == 0 else (x - 1, y) + if color == 1: + return (x + 1, y) if x % 2 == 1 else (x - 1, y) + if color == 2: + return (x, y + 1) if y % 2 == 0 else (x, y - 1) + if color == 3: + return (x, y + 1) if y % 2 == 1 else (x, y - 1) + raise ValueError("matching color must be 0, 1, 2 or 3") + + +def canonical_coordinate_bond(u: Coordinate, v: Coordinate) -> CoordinateBond: + return (u, v) if u < v else (v, u) + + +def representative_bonds(color: int) -> tuple[CoordinateBond, CoordinateBond]: + if color in (0, 1): + start_x = color + return ( + canonical_coordinate_bond((start_x, 0), (start_x + 1, 0)), + canonical_coordinate_bond((start_x, 1), (start_x + 1, 1)), + ) + start_y = color - 2 + return ( + canonical_coordinate_bond((0, start_y), (0, start_y + 1)), + canonical_coordinate_bond((1, start_y), (1, start_y + 1)), + ) + + +def coordinate_heisenberg_bond( + registry: CoordinateRegistry, + bond: CoordinateBond, +) -> PauliSum: + u, v = (registry.site(coordinate) for coordinate in bond) + result = PauliSum.zero() + for op in ("X", "Y", "Z"): + result += PauliSum.term( + PauliString({u: op, v: op}), + Fraction(1, 4), + ) + return result + + +def local_fragment_adjoint( + registry: CoordinateRegistry, + color: int, + operator: PauliSum, +) -> PauliSum: + candidate_bonds: set[CoordinateBond] = set() + for pauli in operator.terms: + for site in pauli.support: + coordinate = registry.coordinate(site) + candidate_bonds.add( + canonical_coordinate_bond( + coordinate, + matching_partner(coordinate, color), + ) + ) + result = PauliSum.zero() + for bond in sorted(candidate_bonds): + result += commutator(coordinate_heisenberg_bond(registry, bond), operator) + return result + + +def local_nested_commutator_density( + key: tuple[int, ...], + cache: dict[tuple[int, ...], tuple[CoordinateRegistry, PauliSum]] | None = None, +) -> tuple[CoordinateRegistry, PauliSum]: + """Return one 2x2-cell density for ``(outermost,...,base)``.""" + + # A fresh registry per complete key keeps cached Pauli site labels valid. + registry = CoordinateRegistry() + base = key[-1] + operator = PauliSum.zero() + for bond in representative_bonds(base): + operator += coordinate_heisenberg_bond(registry, bond) + for color in reversed(key[:-1]): + operator = local_fragment_adjoint(registry, color, operator) + return registry, operator + + +def local_nested_pauli_l1_density(key: tuple[int, ...]) -> Fraction: + _, operator = local_nested_commutator_density(key) + return operator.exact_axis_l1() + + +class LocalDensityEvaluator: + """Suffix-memoized evaluator for connected matching commutators.""" + + def __init__(self) -> None: + self.registries = {color: CoordinateRegistry() for color in range(4)} + self.cache: dict[tuple[int, ...], PauliSum] = {} + + def evaluate(self, key: tuple[int, ...]) -> PauliSum: + if key in self.cache: + return self.cache[key] + base = key[-1] + registry = self.registries[base] + if len(key) == 1: + operator = PauliSum.zero() + for bond in representative_bonds(base): + operator += coordinate_heisenberg_bond(registry, bond) + else: + operator = local_fragment_adjoint( + registry, + key[0], + self.evaluate(key[1:]), + ) + self.cache[key] = operator + return operator + + def pauli_l1_density(self, key: tuple[int, ...]) -> Fraction: + return self.evaluate(key).exact_axis_l1() + + +GaussianInteger = tuple[int, int] +DyadicTerms = dict[PauliString, GaussianInteger] + + +def _multiply_gaussian_by_phase( + coefficient: GaussianInteger, + phase_real: int, + phase_imag: int, +) -> GaussianInteger: + real, imag = coefficient + return ( + real * phase_real - imag * phase_imag, + real * phase_imag + imag * phase_real, + ) + + +def _accumulate_gaussian( + terms: DyadicTerms, + pauli: PauliString, + coefficient: GaussianInteger, +) -> None: + previous = terms.get(pauli) + if previous is None: + if coefficient != (0, 0): + terms[pauli] = coefficient + return + updated = (previous[0] + coefficient[0], previous[1] + coefficient[1]) + if updated == (0, 0): + del terms[pauli] + else: + terms[pauli] = updated + + +def dyadic_local_fragment_adjoint( + registry: CoordinateRegistry, + color: int, + operator: DyadicTerms, +) -> DyadicTerms: + """Apply ``ad_H_color`` to a Gaussian-integer numerator map. + + A Heisenberg bond coefficient is ``1/4`` and a nonzero Pauli + commutator contributes a factor two. Thus every adjoint increases the + shared binary denominator exponent by exactly one. + """ + + candidate_bonds: set[CoordinateBond] = set() + for pauli in operator: + for site in pauli.support: + coordinate = registry.coordinate(site) + candidate_bonds.add( + canonical_coordinate_bond( + coordinate, + matching_partner(coordinate, color), + ) + ) + + result: DyadicTerms = {} + for bond in sorted(candidate_bonds): + u, v = (registry.site(coordinate) for coordinate in bond) + for op in ("X", "Y", "Z"): + bond_pauli = PauliString({u: op, v: op}) + for pauli, coefficient in operator.items(): + if pauli_strings_commute(bond_pauli, pauli): + continue + phase, product_pauli = bond_pauli.multiply(pauli) + _accumulate_gaussian( + result, + product_pauli, + _multiply_gaussian_by_phase( + coefficient, + int(phase.real), + int(phase.imag), + ), + ) + return result + + +class DyadicLocalDensityEvaluator: + """Fast exact evaluator with a shared power-of-two denominator. + + A key of length ``d`` has common coefficient scale ``2**(-(d+1))``. + """ + + def __init__(self) -> None: + self.registries = {color: CoordinateRegistry() for color in range(4)} + self.cache: dict[tuple[int, ...], DyadicTerms] = {} + + @staticmethod + def denominator_exponent(key: tuple[int, ...]) -> int: + return len(key) + 1 + + def evaluate(self, key: tuple[int, ...]) -> DyadicTerms: + if not key: + raise ValueError("nested-commutator key must be nonempty") + if key in self.cache: + return self.cache[key] + base = key[-1] + registry = self.registries[base] + if len(key) == 1: + operator: DyadicTerms = {} + for bond in representative_bonds(base): + u, v = (registry.site(coordinate) for coordinate in bond) + for op in ("X", "Y", "Z"): + _accumulate_gaussian( + operator, + PauliString({u: op, v: op}), + (1, 0), + ) + else: + operator = dyadic_local_fragment_adjoint( + registry, + key[0], + self.evaluate(key[1:]), + ) + self.cache[key] = operator + return operator + + def pauli_l1_density(self, key: tuple[int, ...]) -> Fraction: + numerator = sum( + abs(real) + abs(imag) + for real, imag in self.evaluate(key).values() + ) + return Fraction(numerator, 1 << self.denominator_exponent(key)) + + +SymplecticPauli = tuple[int, int] +SymplecticTerms = dict[SymplecticPauli, int] + + +def _iter_set_bits(mask: int) -> Iterable[int]: + while mask: + least = mask & -mask + yield least.bit_length() - 1 + mask ^= least + + +def _symplectic_product_phase( + left: SymplecticPauli, + right: SymplecticPauli, +) -> tuple[int, SymplecticPauli]: + """Return the exponent ``e`` in ``P(left)P(right)=i**e P(product)``.""" + + left_x, left_z = left + right_x, right_z = right + result = (left_x ^ right_x, left_z ^ right_z) + exponent = ( + (left_x & left_z).bit_count() + + (right_x & right_z).bit_count() + + 2 * (left_z & right_x).bit_count() + - (result[0] & result[1]).bit_count() + ) % 4 + return exponent, result + + +def _symplectic_anticommutes( + left: SymplecticPauli, + right: SymplecticPauli, +) -> bool: + return ( + ((left[0] & right[1]).bit_count() + (left[1] & right[0]).bit_count()) + & 1 + ) == 1 + + +def _symplectic_bond_terms(first: int, second: int) -> tuple[SymplecticPauli, ...]: + sites = (1 << first) | (1 << second) + return ((sites, 0), (sites, sites), (0, sites)) + + +def symplectic_local_fragment_adjoint( + registry: CoordinateRegistry, + color: int, + operator: SymplecticTerms, +) -> SymplecticTerms: + candidate_bonds: set[CoordinateBond] = set() + for x_mask, z_mask in operator: + for site in _iter_set_bits(x_mask | z_mask): + coordinate = registry.coordinate(site) + candidate_bonds.add( + canonical_coordinate_bond( + coordinate, + matching_partner(coordinate, color), + ) + ) + + result: SymplecticTerms = {} + for bond in sorted(candidate_bonds): + first, second = (registry.site(coordinate) for coordinate in bond) + for bond_pauli in _symplectic_bond_terms(first, second): + for pauli, coefficient in operator.items(): + if not _symplectic_anticommutes(bond_pauli, pauli): + continue + phase, product_pauli = _symplectic_product_phase(bond_pauli, pauli) + # Anticommuting Hermitian Paulis have product phase +/- i. + sign = 1 if phase == 1 else -1 + updated = result.get(product_pauli, 0) + sign * coefficient + if updated: + result[product_pauli] = updated + else: + result.pop(product_pauli, None) + return result + + +class SymplecticDyadicLocalDensityEvaluator: + """Fastest exact evaluator: bit-packed Paulis and integer coefficients. + + The common phase ``i**(len(key)-1)`` and denominator + ``2**(len(key)+1)`` are factored out of every coefficient. + """ + + def __init__(self, *, shared_coordinates: bool = False) -> None: + if shared_coordinates: + registry = CoordinateRegistry() + self.registries = {color: registry for color in range(4)} + else: + self.registries = {color: CoordinateRegistry() for color in range(4)} + self.cache: dict[tuple[int, ...], SymplecticTerms] = {} + + @staticmethod + def denominator_exponent(key: tuple[int, ...]) -> int: + return len(key) + 1 + + def evaluate(self, key: tuple[int, ...]) -> SymplecticTerms: + if not key: + raise ValueError("nested-commutator key must be nonempty") + if key in self.cache: + return self.cache[key] + base = key[-1] + registry = self.registries[base] + if len(key) == 1: + operator: SymplecticTerms = {} + for bond in representative_bonds(base): + first, second = (registry.site(coordinate) for coordinate in bond) + for pauli in _symplectic_bond_terms(first, second): + operator[pauli] = operator.get(pauli, 0) + 1 + else: + operator = symplectic_local_fragment_adjoint( + registry, + key[0], + self.evaluate(key[1:]), + ) + self.cache[key] = operator + return operator + + def pauli_l1_density(self, key: tuple[int, ...]) -> Fraction: + numerator = sum(abs(value) for value in self.evaluate(key).values()) + return Fraction(numerator, 1 << self.denominator_exponent(key)) diff --git a/tracks/qcs/solutions/WangTheoPhys/issue128/src/trottercert/matching_screen.py b/tracks/qcs/solutions/WangTheoPhys/issue128/src/trottercert/matching_screen.py new file mode 100644 index 000000000..48e3abd10 --- /dev/null +++ b/tracks/qcs/solutions/WangTheoPhys/issue128/src/trottercert/matching_screen.py @@ -0,0 +1,45 @@ +from __future__ import annotations + +from itertools import permutations + +from .defect_series import right_generator_local_series +from .higher_order import ScalarStage, fourth_order_suzuki_stages + + +MATCHING_ORDERS = tuple(permutations(range(4))) + + +def permutation_from_index(index: int) -> tuple[int, int, int, int]: + if index < 0 or index >= len(MATCHING_ORDERS): + raise IndexError("permutation index must be in [0, 23]") + return MATCHING_ORDERS[index] + + +def screen_matching_order(index: int, order: int = 6) -> dict[str, object]: + if order < 4: + raise ValueError("screening order must be at least four") + permutation = permutation_from_index(index) + stages = tuple( + ScalarStage(permutation[stage.fragment_index], stage.coefficient) + for stage in fourth_order_suzuki_stages(4) + ) + series = right_generator_local_series(stages, order) + metrics = { + str(degree): { + "term_count": len(series[degree]), + "coefficient_l1": float( + sum(abs(value) for value in series[degree].values()) + ), + } + for degree in range(4, order + 1) + } + return { + "schema_version": 1, + "kind": "issue128_matching_order_screen", + "trusted": False, + "purpose": "discovery-ranking", + "permutation_index": index, + "permutation": list(permutation), + "order": order, + "metrics": metrics, + } diff --git a/tracks/qcs/solutions/WangTheoPhys/issue128/src/trottercert/orbits.py b/tracks/qcs/solutions/WangTheoPhys/issue128/src/trottercert/orbits.py new file mode 100644 index 000000000..ecbb0a6b6 --- /dev/null +++ b/tracks/qcs/solutions/WangTheoPhys/issue128/src/trottercert/orbits.py @@ -0,0 +1,137 @@ +from __future__ import annotations + +from dataclasses import dataclass +from typing import Iterable + +from .algebra import PauliString, PauliSum, QComplex +from .lattice import SquareLattice + + +GridOp = tuple[tuple[int, int, str], ...] + + +@dataclass(frozen=True) +class TranslationOrbit: + representative: GridOp + # Coefficient used in one unit-cell density. Translating and summing this + # density reconstructs the global orbit, including finite-size stabilizers. + coefficient: QComplex + members: tuple[PauliString, ...] + unit_cell: tuple[int, int] = (1, 1) + global_coefficient: QComplex = QComplex() + stabilizer: int = 1 + + @property + def width(self) -> int: + return 1 + max(x for x, _, _ in self.representative) + + @property + def height(self) -> int: + return 1 + max(y for _, y, _ in self.representative) + + +def translate_pauli_torus( + pauli: PauliString, + lattice: SquareLattice, + dx: int, + dy: int, +) -> PauliString: + translated: dict[int, str] = {} + for site, op in pauli.ops: + x, y = lattice.coordinates(site) + translated[lattice.site(x + dx, y + dy)] = op + return PauliString(translated) + + +def _grid_form( + pauli: PauliString, + lattice: SquareLattice, + unit_cell: tuple[int, int], +) -> GridOp: + coordinates = [ + (*lattice.coordinates(site), op) + for site, op in pauli.ops + ] + min_x = min(x for x, _, _ in coordinates) + min_y = min(y for _, y, _ in coordinates) + step_x, step_y = unit_cell + shift_x = (min_x // step_x) * step_x + shift_y = (min_y // step_y) * step_y + return tuple( + sorted((x - shift_x, y - shift_y, op) for x, y, op in coordinates) + ) + + +def canonical_translation( + pauli: PauliString, + lattice: SquareLattice, + unit_cell: tuple[int, int] = (1, 1), +) -> tuple[GridOp, tuple[PauliString, ...]]: + if not pauli.ops: + return (), (pauli,) + step_x, step_y = unit_cell + if lattice.length % step_x or lattice.length % step_y: + raise ValueError("unit cell must divide lattice length") + translations = { + translate_pauli_torus(pauli, lattice, dx, dy) + for dx in range(0, lattice.length, step_x) + for dy in range(0, lattice.length, step_y) + } + candidates: list[tuple[tuple[int, int, int, GridOp], GridOp]] = [] + for translated in translations: + grid = _grid_form(translated, lattice, unit_cell) + width = 1 + max(x for x, _, _ in grid) + height = 1 + max(y for _, y, _ in grid) + candidates.append(((width + height, width, height, grid), grid)) + representative = min(candidates)[1] + return representative, tuple(sorted(translations)) + + +def translation_orbits( + operator: PauliSum, + lattice: SquareLattice, + *, + require_invariant: bool = True, + unit_cell: tuple[int, int] = (1, 1), +) -> tuple[TranslationOrbit, ...]: + remaining = set(operator.terms) + result: list[TranslationOrbit] = [] + while remaining: + seed = min(remaining) + representative, members = canonical_translation(seed, lattice, unit_cell) + present = tuple(member for member in members if member in operator.terms) + coefficients = {operator.terms[member] for member in present} + if require_invariant and ( + len(present) != len(members) or len(coefficients) != 1 + ): + raise ValueError("operator is not translation invariant orbit by orbit") + global_coefficient = operator.terms[seed] + step_x, step_y = unit_cell + n_translations = ( + lattice.length // step_x + ) * ( + lattice.length // step_y + ) + if n_translations % len(members): + raise ArithmeticError("translation orbit size does not divide group size") + stabilizer = n_translations // len(members) + coefficient = global_coefficient / stabilizer + result.append( + TranslationOrbit( + representative, + coefficient, + present, + unit_cell, + global_coefficient, + stabilizer, + ) + ) + remaining.difference_update(present) + return tuple(sorted(result, key=lambda orbit: orbit.representative)) + + +def embed_grid_op(grid: GridOp, patch_width: int, offset_x: int, offset_y: int) -> PauliString: + mapping: dict[int, str] = {} + for x, y, op in grid: + mapping[(y + offset_y) * patch_width + x + offset_x] = op + return PauliString(mapping) diff --git a/tracks/qcs/solutions/WangTheoPhys/issue128/src/trottercert/pareto.py b/tracks/qcs/solutions/WangTheoPhys/issue128/src/trottercert/pareto.py new file mode 100644 index 000000000..626093a68 --- /dev/null +++ b/tracks/qcs/solutions/WangTheoPhys/issue128/src/trottercert/pareto.py @@ -0,0 +1,150 @@ +from __future__ import annotations + +from dataclasses import dataclass +from fractions import Fraction +from typing import Sequence + +from .intervals import RationalInterval, nth_root_four_interval, outward_quantize +from .refined_error import defect_tail_site_bound +from .rigorous_fourth import IntervalStage + + +def _second_order_stages( + n_fragments: int, + scale: RationalInterval, +) -> tuple[IntervalStage, ...]: + half = scale / 2 + return ( + *(IntervalStage(index, half) for index in range(n_fragments - 1)), + IntervalStage(n_fragments - 1, scale), + *(IntervalStage(index, half) for index in reversed(range(n_fragments - 1))), + ) + + +def _merge(stages: Sequence[IntervalStage]) -> tuple[IntervalStage, ...]: + merged: list[IntervalStage] = [] + for stage in stages: + if merged and merged[-1].fragment_index == stage.fragment_index: + previous = merged.pop() + merged.append( + IntervalStage( + stage.fragment_index, + previous.coefficient + stage.coefficient, + ) + ) + else: + merged.append(stage) + return tuple(merged) + + +def recursive_suzuki_interval_stages( + order: int, + n_fragments: int = 4, + *, + decimal_digits: int = 12, +) -> tuple[IntervalStage, ...]: + """Return the standard five-copy Suzuki formula of even ``order``.""" + + if order < 2 or order % 2: + raise ValueError("Suzuki order must be a positive even integer") + stages = _second_order_stages(n_fragments, RationalInterval.point(1)) + current_order = 2 + grid = 10**decimal_digits + while current_order < order: + next_order = current_order + 2 + root = nth_root_four_interval(next_order - 1, decimal_digits) + scale = outward_quantize( + RationalInterval.point(1) / (4 - root), + grid, + ) + copy_scales = (scale, scale, 1 - 4 * scale, scale, scale) + composed: list[IntervalStage] = [] + for copy_scale in copy_scales: + composed.extend( + IntervalStage( + stage.fragment_index, + stage.coefficient * copy_scale, + ) + for stage in stages + ) + stages = _merge(composed) + current_order = next_order + return stages + + +@dataclass(frozen=True) +class PublishedSuzukiPoint: + order: int + steps: int + stages_per_step: int + group_exponentials: int + global_error_bound: Fraction + + +def published_suzuki_tail_bound( + stages: Sequence[IntervalStage], + *, + order: int, + n_sites: int, + steps: int, +) -> Fraction: + """Rigorous generic locality bound using only the published formula order.""" + + return Fraction(n_sites) * defect_tail_site_bound( + stages, + steps, + first_omitted_degree=order, + ) + + +def minimum_published_suzuki_point( + order: int, + *, + n_sites: int, + tolerance: Fraction, + n_fragments: int = 4, + decimal_digits: int = 12, +) -> PublishedSuzukiPoint: + stages = recursive_suzuki_interval_stages( + order, + n_fragments, + decimal_digits=decimal_digits, + ) + + def bound(steps: int) -> Fraction: + return published_suzuki_tail_bound( + stages, + order=order, + n_sites=n_sites, + steps=steps, + ) + + low = 1 + high = 1 + while True: + try: + accepted = bound(high) <= tolerance + except ValueError: + accepted = False + if accepted: + break + high *= 2 + while low < high: + middle = (low + high) // 2 + try: + accepted = bound(middle) <= tolerance + except ValueError: + accepted = False + if accepted: + high = middle + else: + low = middle + 1 + + groups = (len(stages) - 1) * low + 1 + return PublishedSuzukiPoint( + order=order, + steps=low, + stages_per_step=len(stages), + group_exponentials=groups, + global_error_bound=bound(low), + ) diff --git a/tracks/qcs/solutions/WangTheoPhys/issue128/src/trottercert/processors.py b/tracks/qcs/solutions/WangTheoPhys/issue128/src/trottercert/processors.py new file mode 100644 index 000000000..a030c43af --- /dev/null +++ b/tracks/qcs/solutions/WangTheoPhys/issue128/src/trottercert/processors.py @@ -0,0 +1,166 @@ +from __future__ import annotations + +from dataclasses import dataclass +from fractions import Fraction +from math import sqrt +from typing import Sequence + +import sympy as sp + +from .algebra import PauliString, PauliSum, QComplex, commutator +from .hamiltonian import full_heisenberg_hamiltonian, heisenberg_bond +from .lattice import SquareLattice + + +def spin_chirality(u: int, v: int, w: int) -> PauliSum: + """Return chi(u,v,w) with [h_uv,h_vw] = -i chi(u,v,w).""" + + return commutator(heisenberg_bond(u, v), heisenberg_bond(v, w)).scale( + QComplex(0, 1) + ) + + +def global_chirality_color_basis( + lattice: SquareLattice, +) -> tuple[tuple[tuple[int, int], PauliSum], ...]: + """Sums of oriented path chiralities grouped by matching colors. + + Only color pairs ``a < b`` are retained because reversing the path + orientation changes the sign. + """ + + matchings = lattice.four_matchings() + neighbor: dict[tuple[int, int], int] = {} + for color, edges in enumerate(matchings): + for u, v in edges: + neighbor[(u, color)] = v + neighbor[(v, color)] = u + result: list[tuple[tuple[int, int], PauliSum]] = [] + for first in range(4): + for second in range(first + 1, 4): + operator = PauliSum.zero() + for center in range(lattice.n_sites): + operator += spin_chirality( + neighbor[(center, first)], + center, + neighbor[(center, second)], + ) + result.append(((first, second), operator)) + return tuple(result) + + +def processor_images( + hamiltonian: PauliSum, + basis: Sequence[PauliSum], +) -> tuple[PauliSum, ...]: + """Return i[Q_a,H], a Hermitian operator for every Hermitian Q_a.""" + + images = tuple( + commutator(operator, hamiltonian).scale(QComplex(0, 1)) + for operator in basis + ) + if not all(image.is_hermitian() for image in images): + raise ArithmeticError("processor image is not Hermitian") + return images + + +@dataclass(frozen=True) +class ProcessorObstruction: + coefficients: tuple[Fraction, ...] + residual: PauliSum + target_l2: float + residual_l2: float + exact_solution: bool + + @property + def relative_l2(self) -> float: + return self.residual_l2 / self.target_l2 if self.target_l2 else 0.0 + + +def _real_coefficient(operator: PauliSum, pauli: PauliString) -> Fraction: + coefficient = operator.terms.get(pauli, QComplex()) + if coefficient.imag: + raise ValueError("processor projection requires Hermitian real Pauli coefficients") + return coefficient.real + + +def solve_processor_obstruction( + target: PauliSum, + images: Sequence[PauliSum], +) -> ProcessorObstruction: + """Project ``-target`` onto the exact rational span of processor images. + + The projection metric is the Hilbert--Schmidt/Pauli coefficient + Euclidean metric. It is used only to diagnose the algebraic obstruction, + not as an operator-norm certificate. + """ + + paulis = sorted( + set(target.terms).union(*(set(image.terms) for image in images)) + ) + columns = len(images) + gram = [[Fraction() for _ in range(columns)] for _ in range(columns)] + rhs = [Fraction() for _ in range(columns)] + for pauli in paulis: + row = [_real_coefficient(image, pauli) for image in images] + target_value = _real_coefficient(target, pauli) + for left in range(columns): + rhs[left] -= row[left] * target_value + for right in range(columns): + gram[left][right] += row[left] * row[right] + + matrix = sp.Matrix( + [[sp.Rational(value.numerator, value.denominator) for value in row] for row in gram] + ) + vector = sp.Matrix( + [sp.Rational(value.numerator, value.denominator) for value in rhs] + ) + solution_set = sp.linsolve((matrix, vector)) + solution_tuple = next(iter(solution_set), ()) + if not solution_tuple: + coefficients = tuple(Fraction() for _ in images) + else: + substitutions = { + symbol: sp.Integer(0) + for expression in solution_tuple + for symbol in expression.free_symbols + } + coefficients = tuple( + Fraction(int(value.p), int(value.q)) + for expression in solution_tuple + for value in [sp.cancel(expression.subs(substitutions))] + ) + + residual = target.copy() + for coefficient, image in zip(coefficients, images): + residual += image.scale(coefficient) + + def squared_l2(operator: PauliSum) -> Fraction: + total = Fraction() + for coefficient in operator.terms.values(): + if coefficient.imag: + total += coefficient.real**2 + coefficient.imag**2 + else: + total += coefficient.real**2 + return total + + target_l2 = sqrt(float(squared_l2(target))) + residual_l2 = sqrt(float(squared_l2(residual))) + return ProcessorObstruction( + coefficients=coefficients, + residual=residual, + target_l2=target_l2, + residual_l2=residual_l2, + exact_solution=not residual, + ) + + +def heisenberg_color_processor_obstruction( + lattice: SquareLattice, + target: PauliSum, +) -> tuple[tuple[tuple[int, int], ...], ProcessorObstruction]: + labeled = global_chirality_color_basis(lattice) + labels = tuple(label for label, _ in labeled) + basis = tuple(operator for _, operator in labeled) + images = processor_images(full_heisenberg_hamiltonian(lattice), basis) + return labels, solve_processor_obstruction(target, images) diff --git a/tracks/qcs/solutions/WangTheoPhys/issue128/src/trottercert/refined_error.py b/tracks/qcs/solutions/WangTheoPhys/issue128/src/trottercert/refined_error.py new file mode 100644 index 000000000..da2f7a5d7 --- /dev/null +++ b/tracks/qcs/solutions/WangTheoPhys/issue128/src/trottercert/refined_error.py @@ -0,0 +1,425 @@ +from __future__ import annotations + +from dataclasses import dataclass +from fractions import Fraction +from math import factorial +from typing import Sequence + +from .intervals import RationalInterval, outward_quantize +from .local_commutators import ( + CoordinateRegistry, + SymplecticDyadicLocalDensityEvaluator, + SymplecticPauli, +) +from .rigorous_fourth import ( + IntervalStage, + fourth_order_suzuki_interval_stages, +) + + +IntervalWord = tuple[int, ...] +IntervalWordSeries = list[dict[IntervalWord, RationalInterval]] + +# For h=(XX+YY+ZZ)/4 and any phase-free Pauli string P, either zero or +# exactly two of XX, YY, ZZ anticommute with P. Each nonzero commutator +# contributes l1 weight 2/4, hence ||[h,P]||_{P,1} <= 1. +HEISENBERG_BOND_PAULI_L1_GROWTH = Fraction(1) + + +def symplectic_pauli_from_coordinates( + registry: CoordinateRegistry, + coordinates: Sequence[tuple[int, int, str]], +) -> SymplecticPauli: + x_mask = z_mask = 0 + for x, y, op in coordinates: + if op not in {"X", "Y", "Z"}: + raise ValueError(f"unsupported Pauli axis {op!r}") + bit = 1 << registry.site((x, y)) + if op in {"X", "Y"}: + x_mask |= bit + if op in {"Z", "Y"}: + z_mask |= bit + return x_mask, z_mask + + +def canonicalize_symplectic_unit_cell( + registry: CoordinateRegistry, + pauli: SymplecticPauli, + unit_cell: tuple[int, int] = (2, 2), +) -> SymplecticPauli: + step_x, step_y = unit_cell + if step_x < 1 or step_y < 1: + raise ValueError("unit-cell dimensions must be positive") + x_mask, z_mask = pauli + sites = x_mask | z_mask + coordinates: list[tuple[int, int, str]] = [] + while sites: + bit = sites & -sites + site = bit.bit_length() - 1 + x = bool(x_mask & bit) + z = bool(z_mask & bit) + op = "Y" if x and z else ("X" if x else "Z") + coordinate_x, coordinate_y = registry.coordinate(site) + coordinates.append((coordinate_x, coordinate_y, op)) + sites ^= bit + if not coordinates: + return 0, 0 + min_x = min(x for x, _, _ in coordinates) + min_y = min(y for _, y, _ in coordinates) + shift_x = -(min_x - min_x % step_x) + shift_y = -(min_y - min_y % step_y) + return symplectic_pauli_from_coordinates( + registry, + tuple( + (x + shift_x, y + shift_y, op) + for x, y, op in coordinates + ), + ) + + +def _interval_series_identity(order: int) -> IntervalWordSeries: + result = [{} for _ in range(order + 1)] + result[0][()] = RationalInterval.point(1) + return result + + +def _interval_series_add( + left: IntervalWordSeries, + right: IntervalWordSeries, +) -> IntervalWordSeries: + result: IntervalWordSeries = [] + zero = RationalInterval.point(0) + for left_degree, right_degree in zip(left, right): + terms = dict(left_degree) + for word, coefficient in right_degree.items(): + terms[word] = terms.get(word, zero) + coefficient + result.append(terms) + return result + + +def _interval_series_scale( + series: IntervalWordSeries, + scalar: Fraction, +) -> IntervalWordSeries: + return [ + {word: coefficient * scalar for word, coefficient in degree.items()} + for degree in series + ] + + +def _interval_series_multiply( + left: IntervalWordSeries, + right: IntervalWordSeries, +) -> IntervalWordSeries: + order = len(left) - 1 + result: IntervalWordSeries = [{} for _ in range(order + 1)] + zero = RationalInterval.point(0) + for degree in range(order + 1): + terms = result[degree] + for left_degree in range(degree + 1): + for left_word, left_coefficient in left[left_degree].items(): + for right_word, right_coefficient in right[degree - left_degree].items(): + word = left_word + right_word + terms[word] = terms.get(word, zero) + ( + left_coefficient * right_coefficient + ) + return result + + +def _interval_exponential( + fragment_index: int, + coefficient: RationalInterval, + order: int, +) -> IntervalWordSeries: + result: IntervalWordSeries = [{} for _ in range(order + 1)] + for degree in range(order + 1): + result[degree][(fragment_index,) * degree] = ( + coefficient**degree / factorial(degree) + ) + return result + + +def interval_formula_log_series( + stages: Sequence[IntervalStage], + order: int, +) -> IntervalWordSeries: + product = _interval_series_identity(order) + for stage in stages: + product = _interval_series_multiply( + product, + _interval_exponential( + stage.fragment_index, + stage.coefficient, + order, + ), + ) + delta = _interval_series_add( + product, + _interval_series_scale(_interval_series_identity(order), Fraction(-1)), + ) + logarithm: IntervalWordSeries = [{} for _ in range(order + 1)] + power = _interval_series_identity(order) + for exponent in range(1, order + 1): + power = _interval_series_multiply(power, delta) + logarithm = _interval_series_add( + logarithm, + _interval_series_scale( + power, + Fraction(1 if exponent % 2 else -1, exponent), + ), + ) + return logarithm + + +def _leading_e5_cell_interval_coefficients( + stages: Sequence[IntervalStage], + *, + quantization_digits: int, + canonicalize: bool, +) -> dict[SymplecticPauli, RationalInterval]: + logarithm = interval_formula_log_series(stages, 5) + grid = 10**quantization_digits + word_endpoints: dict[IntervalWord, tuple[int, int]] = {} + for word, coefficient in logarithm[5].items(): + rounded = outward_quantize(coefficient, grid) + word_endpoints[word] = ( + rounded.lower.numerator * (grid // rounded.lower.denominator), + rounded.upper.numerator * (grid // rounded.upper.denominator), + ) + + evaluator = SymplecticDyadicLocalDensityEvaluator(shared_coordinates=True) + registry = evaluator.registries[0] + coefficients: dict[SymplecticPauli, tuple[int, int]] = {} + for word, (word_lower, word_upper) in word_endpoints.items(): + for raw_pauli, numerator in evaluator.evaluate(word).items(): + pauli = ( + canonicalize_symplectic_unit_cell(registry, raw_pauli) + if canonicalize + else raw_pauli + ) + lower, upper = coefficients.get(pauli, (0, 0)) + if numerator >= 0: + lower += numerator * word_lower + upper += numerator * word_upper + else: + lower += numerator * word_upper + upper += numerator * word_lower + if lower or upper: + coefficients[pauli] = (lower, upper) + else: + coefficients.pop(pauli, None) + # Dynkin projection divides by degree five; local depth-five coefficients + # have common dyadic denominator 2**6. + denominator = grid * 5 * (1 << 6) + return { + pauli: RationalInterval( + Fraction(lower, denominator), + Fraction(upper, denominator), + ) + for pauli, (lower, upper) in coefficients.items() + } + + +def certified_leading_e5_cell_l1( + stages: Sequence[IntervalStage], + *, + quantization_digits: int = 24, +) -> Fraction: + coefficients = _leading_e5_cell_interval_coefficients( + stages, + quantization_digits=quantization_digits, + canonicalize=False, + ) + return sum( + (coefficient.abs_upper() for coefficient in coefficients.values()), + Fraction(), + ) + + +def certified_d4_cell_coefficients( + stages: Sequence[IntervalStage], + *, + quantization_digits: int = 18, +) -> dict[SymplecticPauli, RationalInterval]: + return { + pauli: coefficient * 5 + for pauli, coefficient in _leading_e5_cell_interval_coefficients( + stages, + quantization_digits=quantization_digits, + canonicalize=True, + ).items() + } + + +def certified_e7_cell_l1_majorant( + stages: Sequence[IntervalStage], +) -> Fraction: + logarithm = interval_formula_log_series(stages, 7) + word_l1 = sum( + (coefficient.abs_upper() for coefficient in logarithm[7].values()), + Fraction(), + ) + # A length-seven right-nested commutator starts from the 3/2 l1 cell + # density and gains at most s at support size s=2,...,7. A matching + # contains at most one overlapping bond per support site, and one bond + # has exact Pauli-l1 growth constant one. + nested_max = Fraction(3, 2) + for support_size in range(2, 8): + nested_max *= ( + HEISENBERG_BOND_PAULI_L1_GROWTH * support_size + ) + return word_l1 * nested_max / 7 + + +def _abs_upper(interval: RationalInterval) -> Fraction: + return interval.abs_upper() + + +def defect_tail_site_bound( + stages: Sequence[IntervalStage], + steps: int, + *, + first_omitted_degree: int = 8, +) -> Fraction: + """Total-time, per-site tail bound for right-generator degrees >= q. + + A cell starts with Pauli-l1 density 3/2. After ``q`` local adjoints, + support counting gives at most + + (3/2) * (q+1)! + + per cell. The conjugation Taylor coefficient contributes ``1/q!``; + the Duhamel time integral contributes ``1/(q+1)``. Those factorial + factors cancel, leaving the geometric series summed below. Dividing + the four-site cell density by four produces the prefactor 3/8. + """ + + prefix = Fraction() + total = Fraction() + for stage in stages: + coefficient = _abs_upper(stage.coefficient) + ratio = ( + HEISENBERG_BOND_PAULI_L1_GROWTH * prefix / steps + ) + if ratio >= 1: + raise ValueError("step count is outside the convergence region") + total += coefficient * ratio**first_omitted_degree / (1 - ratio) + prefix += coefficient + # Cell base density is 3/2; divide by four sites per cell. + return Fraction(3, 8) * total + + +@dataclass(frozen=True) +class RefinedFourthOrderBound: + steps: int + e5_site_l1: Fraction + e7_site_majorant: Fraction + degree_four_contribution: Fraction + degree_five_contribution: Fraction + degree_six_contribution: Fraction + degree_seven_contribution: Fraction + tail_contribution: Fraction + global_error_bound: Fraction + + +@dataclass(frozen=True) +class RefinedFourthOrderConstants: + stages: tuple[IntervalStage, ...] + e5_site_l1: Fraction + e7_site_majorant: Fraction + d4_site: Fraction + d5_site: Fraction + d6_site: Fraction + d7_site: Fraction + + +def build_refined_fourth_order_constants( + *, + decimal_digits: int = 12, + quantization_digits: int = 18, +) -> RefinedFourthOrderConstants: + stages, _ = fourth_order_suzuki_interval_stages( + 4, + decimal_digits=decimal_digits, + ) + e5_site = certified_leading_e5_cell_l1( + stages, + quantization_digits=quantization_digits, + ) / 4 + e7_site = certified_e7_cell_l1_majorant(stages) / 4 + + # ad_H on an s-site Pauli term has at most 4s overlapping square-lattice + # bonds. The exact single-bond Pauli-l1 growth constant is one. + h_e5 = 24 * e5_site + hh_e5 = 28 * h_e5 + hhh_e5 = 32 * hh_e5 + h_e7 = 32 * e7_site + + return RefinedFourthOrderConstants( + stages=tuple(stages), + e5_site_l1=e5_site, + e7_site_majorant=e7_site, + d4_site=5 * e5_site, + d5_site=2 * h_e5, + d6_site=7 * e7_site + Fraction(2, 3) * hh_e5, + d7_site=3 * h_e7 + Fraction(1, 6) * hhh_e5, + ) + + +def evaluate_refined_fourth_order_bound( + constants: RefinedFourthOrderConstants, + n_sites: int, + steps: int, + *, + d4_site_override: Fraction | None = None, +) -> RefinedFourthOrderBound: + if d4_site_override is not None: + if d4_site_override < 0: + raise ValueError("D4 site-density override must be nonnegative") + if d4_site_override > constants.d4_site: + raise ValueError( + "D4 site-density override cannot exceed the existing bound" + ) + + def contribution(degree: int, density: Fraction) -> Fraction: + return Fraction(n_sites) * density / ( + (degree + 1) * steps**degree + ) + + c4 = contribution( + 4, + ( + constants.d4_site + if d4_site_override is None + else d4_site_override + ), + ) + c5 = contribution(5, constants.d5_site) + c6 = contribution(6, constants.d6_site) + c7 = contribution(7, constants.d7_site) + tail = Fraction(n_sites) * defect_tail_site_bound(constants.stages, steps) + return RefinedFourthOrderBound( + steps=steps, + e5_site_l1=constants.e5_site_l1, + e7_site_majorant=constants.e7_site_majorant, + degree_four_contribution=c4, + degree_five_contribution=c5, + degree_six_contribution=c6, + degree_seven_contribution=c7, + tail_contribution=tail, + global_error_bound=c4 + c5 + c6 + c7 + tail, + ) + + +def refined_fourth_order_bound( + n_sites: int, + steps: int, + *, + decimal_digits: int = 12, + quantization_digits: int = 18, +) -> RefinedFourthOrderBound: + constants = build_refined_fourth_order_constants( + decimal_digits=decimal_digits, + quantization_digits=quantization_digits, + ) + return evaluate_refined_fourth_order_bound(constants, n_sites, steps) diff --git a/tracks/qcs/solutions/WangTheoPhys/issue128/src/trottercert/representation.py b/tracks/qcs/solutions/WangTheoPhys/issue128/src/trottercert/representation.py new file mode 100644 index 000000000..a6f199885 --- /dev/null +++ b/tracks/qcs/solutions/WangTheoPhys/issue128/src/trottercert/representation.py @@ -0,0 +1,36 @@ +from __future__ import annotations + +from itertools import product + +import sympy as sp + +from .algebra import commutator +from .hamiltonian import four_matching_fragments +from .lattice import SquareLattice + + +def degree_three_four_matching_rank(length: int = 4) -> tuple[int, int, int]: + """Exact rational rank of all ``[Ha,[Hb,Hc]]`` operator images.""" + + lattice = SquareLattice(length) + fragments = four_matching_fragments(lattice) + keys = tuple(product(range(4), repeat=3)) + operators = tuple( + commutator( + fragments[first], + commutator(fragments[second], fragments[third]), + ) + for first, second, third in keys + ) + paulis = tuple(sorted(set().union(*(operator.terms for operator in operators)))) + row = {pauli: index for index, pauli in enumerate(paulis)} + matrix = sp.MutableSparseMatrix(len(paulis), len(keys), {}) + for column, operator in enumerate(operators): + for pauli, coefficient in operator.terms.items(): + if coefficient.imag: + raise ArithmeticError("degree-three image should be Hermitian") + matrix[row[pauli], column] = sp.Rational( + coefficient.real.numerator, + coefficient.real.denominator, + ) + return int(matrix.rank()), len(keys), len(paulis) diff --git a/tracks/qcs/solutions/WangTheoPhys/issue128/src/trottercert/resources.py b/tracks/qcs/solutions/WangTheoPhys/issue128/src/trottercert/resources.py new file mode 100644 index 000000000..36a7ecc71 --- /dev/null +++ b/tracks/qcs/solutions/WangTheoPhys/issue128/src/trottercert/resources.py @@ -0,0 +1,83 @@ +from __future__ import annotations + +from dataclasses import dataclass +from fractions import Fraction + + +def ceil_nth_root_fraction(value: Fraction, degree: int) -> int: + if value < 0 or degree <= 0: + raise ValueError("requires a nonnegative value and positive degree") + if value == 0: + return 0 + low, high = 0, 1 + while high**degree * value.denominator < value.numerator: + high *= 2 + while low + 1 < high: + middle = (low + high) // 2 + if middle**degree * value.denominator >= value.numerator: + high = middle + else: + low = middle + return high + + +def required_steps( + error_constant: Fraction, + tolerance: Fraction, + formula_order: int, + time: Fraction = Fraction(1), +) -> int: + target = error_constant * time ** (formula_order + 1) / tolerance + return ceil_nth_root_fraction(target, formula_order) + + +def symmetric_group_exponentials(n_fragments: int, steps: int) -> int: + if n_fragments < 2 or steps < 1: + raise ValueError("requires at least two fragments and one step") + return (2 * n_fragments - 2) * steps + 1 + + +@dataclass(frozen=True) +class ResourceEstimate: + steps: int + group_exponentials: int + local_propagators: int + cnot_upper: int + + +def four_matching_resources( + n_sites: int, + steps: int, + *, + cnots_per_bond: int = 3, +) -> ResourceEstimate: + groups = symmetric_group_exponentials(4, steps) + local = groups * (n_sites // 2) + return ResourceEstimate(steps, groups, local, local * cnots_per_bond) + + +def three_l_path_resources( + n_sites: int, + steps: int, + *, + cnots_per_cluster: int = 9, +) -> ResourceEstimate: + groups = symmetric_group_exponentials(3, steps) + local = groups * (n_sites // 3) + return ResourceEstimate(steps, groups, local, local * cnots_per_cluster) + + +def fourth_order_four_matching_resources( + n_sites: int, + steps: int, + *, + stage_count: int = 31, + cnots_per_bond: int = 3, +) -> ResourceEstimate: + """Resources for a palindromic S4 macro-step with mergeable boundaries.""" + + if stage_count < 1 or steps < 1: + raise ValueError("stage_count and steps must be positive") + groups = (stage_count - 1) * steps + 1 + local = groups * (n_sites // 2) + return ResourceEstimate(steps, groups, local, local * cnots_per_bond) diff --git a/tracks/qcs/solutions/WangTheoPhys/issue128/src/trottercert/rigorous_fourth.py b/tracks/qcs/solutions/WangTheoPhys/issue128/src/trottercert/rigorous_fourth.py new file mode 100644 index 000000000..cbed197d9 --- /dev/null +++ b/tracks/qcs/solutions/WangTheoPhys/issue128/src/trottercert/rigorous_fourth.py @@ -0,0 +1,424 @@ +from __future__ import annotations + +from dataclasses import dataclass +from fractions import Fraction +from math import factorial, lcm +from typing import Sequence + +import numpy as np + +from .higher_order import _multinomial, weak_compositions +from .intervals import ( + RationalInterval, + cube_root_four_interval, + outward_quantize, +) +from .local_commutators import ( + SymplecticDyadicLocalDensityEvaluator, + SymplecticPauli, +) + + +@dataclass(frozen=True, slots=True) +class IntervalStage: + fragment_index: int + coefficient: RationalInterval + + +@dataclass(frozen=True) +class FourthOrderRationalCertificate: + center: int + root_interval: RationalInterval + cell_density_upper: Fraction + site_density_upper: Fraction + theorem_terms: int + paired_terms: int + singleton_terms: int + + +@dataclass(frozen=True) +class FourthOrderPublishedTriangleCertificate: + center: int + root_interval: RationalInterval + cell_density_upper: Fraction + site_density_upper: Fraction + theorem_terms: int + expanded_commutator_keys: int + + +def _second_order_interval_stages( + n_fragments: int, + scale: RationalInterval, +) -> list[IntervalStage]: + half = scale / 2 + stages = [ + IntervalStage(index, half) + for index in range(n_fragments - 1) + ] + stages.append(IntervalStage(n_fragments - 1, scale)) + stages.extend( + IntervalStage(index, half) + for index in reversed(range(n_fragments - 1)) + ) + return stages + + +def _merge_interval_stages( + stages: Sequence[IntervalStage], +) -> tuple[IntervalStage, ...]: + merged: list[IntervalStage] = [] + for stage in stages: + if merged and merged[-1].fragment_index == stage.fragment_index: + previous = merged.pop() + merged.append( + IntervalStage( + stage.fragment_index, + previous.coefficient + stage.coefficient, + ) + ) + else: + merged.append(stage) + return tuple(merged) + + +def fourth_order_suzuki_interval_stages( + n_fragments: int = 4, + *, + decimal_digits: int = 24, +) -> tuple[tuple[IntervalStage, ...], RationalInterval]: + root = cube_root_four_interval(decimal_digits) + grid = 10**decimal_digits + u = outward_quantize( + RationalInterval.point(1) / (4 - root), + grid, + ) + scales = (u, u, 1 - 4 * u, u, u) + stages: list[IntervalStage] = [] + for scale in scales: + stages.extend(_second_order_interval_stages(n_fragments, scale)) + return _merge_interval_stages(stages), root + + +def _anticommutes(left: SymplecticPauli, right: SymplecticPauli) -> bool: + return ( + ((left[0] & right[1]).bit_count() + (left[1] & right[0]).bit_count()) + & 1 + ) == 1 + + +def _greedy_pairs( + paulis: Sequence[SymplecticPauli], + midpoint_coefficients: Sequence[int | Fraction], +) -> tuple[tuple[tuple[int, int], ...], tuple[int, ...]]: + # Floating point is used only to choose a partition. Soundness does not + # depend on that choice: every returned pair is checked exactly below. + magnitudes = np.fromiter( + (abs(float(value)) for value in midpoint_coefficients), + dtype=np.float64, + count=len(midpoint_coefficients), + ) + order = np.argsort(-magnitudes, kind="stable") + used = np.zeros(len(paulis), dtype=np.bool_) + pairs: list[tuple[int, int]] = [] + singles: list[int] = [] + for position, index_value in enumerate(order): + index = int(index_value) + if used[index] or midpoint_coefficients[index] == 0: + continue + used[index] = True + partner = None + for candidate_value in order[position + 1 :]: + candidate = int(candidate_value) + if used[candidate] or midpoint_coefficients[candidate] == 0: + continue + if _anticommutes(paulis[index], paulis[candidate]): + partner = candidate + break + if partner is None: + singles.append(index) + else: + used[partner] = True + pairs.append((index, partner)) + # Exactly-zero midpoint intervals are harmless singletons; including + # them is necessary when a narrow interval still contains nonzero values. + for index_value in order: + index = int(index_value) + if not used[index]: + used[index] = True + singles.append(index) + return tuple(pairs), tuple(singles) + + +def _rational_pair_bound( + coefficients: Sequence[RationalInterval], + pairs: Sequence[tuple[int, int]], + singles: Sequence[int], +) -> Fraction: + total = sum((coefficients[index].abs_upper() for index in singles), Fraction()) + for first, second in pairs: + left = coefficients[first] + right = coefficients[second] + # For a,b >= 0: + # sqrt(a^2+b^2) <= a+b-(1/2)min(a,b). + total += ( + left.abs_upper() + + right.abs_upper() + - min(left.abs_lower(), right.abs_lower()) / 2 + ) + return total + + +def _fixed_base_intervals( + base: Sequence[RationalInterval], + dyadic_denominator: int, +) -> tuple[int, tuple[tuple[int, int], ...]]: + common = 1 + for interval in base: + common = lcm( + common, + interval.lower.denominator, + interval.upper.denominator, + ) + endpoints = tuple( + ( + interval.lower.numerator * (common // interval.lower.denominator), + interval.upper.numerator * (common // interval.upper.denominator), + ) + for interval in base + ) + return common * dyadic_denominator, endpoints + + +def _fixed_linear_interval( + numerators: Sequence[int], + base_endpoints: Sequence[tuple[int, int]], +) -> tuple[int, int]: + lower = upper = 0 + for numerator, (base_lower, base_upper) in zip(numerators, base_endpoints): + if numerator >= 0: + lower += numerator * base_lower + upper += numerator * base_upper + else: + lower += numerator * base_upper + upper += numerator * base_lower + return lower, upper + + +def _fixed_abs_bounds(interval: tuple[int, int]) -> tuple[int, int]: + lower, upper = interval + absolute_upper = max(abs(lower), abs(upper)) + if lower <= 0 <= upper: + return 0, absolute_upper + return min(abs(lower), abs(upper)), absolute_upper + + +def _fixed_pair_bound_numerator( + intervals: Sequence[tuple[int, int]], + pairs: Sequence[tuple[int, int]], + singles: Sequence[int], +) -> int: + """Return the pair bound numerator over twice the common denominator.""" + + absolute = tuple(_fixed_abs_bounds(interval) for interval in intervals) + total = sum(2 * absolute[index][1] for index in singles) + for first, second in pairs: + first_lower, first_upper = absolute[first] + second_lower, second_upper = absolute[second] + total += ( + 2 * first_upper + + 2 * second_upper + - min(first_lower, second_lower) + ) + return total + + +def fourth_order_rational_pair_certificate( + *, + center: int = 17, + decimal_digits: int = 24, +) -> FourthOrderRationalCertificate: + """Build a fully rational S4 local-norm certificate.""" + + stages_left, root = fourth_order_suzuki_interval_stages( + 4, + decimal_digits=decimal_digits, + ) + stages = tuple(reversed(stages_left)) + count = len(stages) + if not 1 <= center <= count: + raise ValueError("center must use one-based stage indexing") + order = 4 + grouped: dict[ + tuple[tuple[int, ...], int], + Fraction, + ] = {} + base_by_j: dict[int, tuple[RationalInterval, ...]] = {} + theorem_terms = 0 + + def collect( + j: int, + indices: tuple[int, ...], + composition: tuple[int, ...], + ) -> None: + nonlocal theorem_terms + theorem_terms += 1 + outer: list[int] = [] + scalar = RationalInterval.point(_multinomial(order, composition)) + for stage_index, power in zip(indices, composition): + stage = stages[stage_index - 1] + scalar *= stage.coefficient**power + outer.extend([stage.fragment_index] * power) + if j not in base_by_j: + base = [RationalInterval.point(0) for _ in range(4)] + for base_index in range(1, j): + stage = stages[base_index - 1] + base[stage.fragment_index] += stage.coefficient + base_by_j[j] = tuple(base) + key = (tuple(outer), j) + grouped[key] = grouped.get(key, Fraction()) + scalar.abs_upper() + + for j in range(2, center + 1): + indices = tuple(range(center, j - 1, -1)) + for composition in weak_compositions(order, len(indices)): + if composition[-1]: + collect(j, indices, composition) + for j in range(center + 1, count + 1): + indices = tuple(range(center + 1, j + 1)) + for composition in weak_compositions(order, len(indices)): + if composition[-1]: + collect(j, indices, composition) + + evaluator = SymplecticDyadicLocalDensityEvaluator(shared_coordinates=True) + denominator = 1 << (order + 2) + fixed_base = { + j: _fixed_base_intervals(base, denominator) + for j, base in base_by_j.items() + } + coefficient_cache: dict[ + tuple[int, tuple[int, int, int, int]], + tuple[int, int], + ] = {} + total = Fraction() + paired_terms = 0 + singleton_terms = 0 + for (outer, j), scalar_upper in grouped.items(): + operators = [evaluator.evaluate(outer + (base,)) for base in range(4)] + paulis = tuple(sorted(set().union(*(operator.keys() for operator in operators)))) + common_denominator, base_endpoints = fixed_base[j] + coefficients: list[tuple[int, int]] = [] + for pauli in paulis: + numerators = tuple( + operator.get(pauli, 0) + for operator in operators + ) + cache_key = (j, numerators) + coefficient = coefficient_cache.get(cache_key) + if coefficient is None: + coefficient = _fixed_linear_interval( + numerators, + base_endpoints, + ) + coefficient_cache[cache_key] = coefficient + coefficients.append(coefficient) + midpoint = tuple( + coefficient[0] + coefficient[1] + for coefficient in coefficients + ) + pairs, singles = _greedy_pairs(paulis, midpoint) + # The verifier checks every selected pair, rather than trusting the + # discovery ordering. + if any(not _anticommutes(paulis[a], paulis[b]) for a, b in pairs): + raise ArithmeticError("invalid anticommuting pair") + local_bound = Fraction( + _fixed_pair_bound_numerator(coefficients, pairs, singles), + 2 * common_denominator, + ) + total += scalar_upper * local_bound + paired_terms += len(pairs) + singleton_terms += len(singles) + + cell = total / factorial(order + 1) + return FourthOrderRationalCertificate( + center=center, + root_interval=root, + cell_density_upper=cell, + site_density_upper=cell / 4, + theorem_terms=theorem_terms, + paired_terms=paired_terms, + singleton_terms=singleton_terms, + ) + + +def fourth_order_published_triangle_certificate( + *, + center: int = 20, + decimal_digits: int = 18, +) -> FourthOrderPublishedTriangleCertificate: + """Instantiate the published high-order theorem with full triangle expansion. + + This intentionally expands every partial Hamiltonian sum before taking + local Pauli-l1 norms. It is the independently reproducible published + baseline, not the strengthened candidate analysis. + """ + + stages_left, root = fourth_order_suzuki_interval_stages( + 4, + decimal_digits=decimal_digits, + ) + stages = tuple(reversed(stages_left)) + count = len(stages) + if not 1 <= center <= count: + raise ValueError("center must use one-based stage indexing") + order = 4 + weights: dict[tuple[int, ...], Fraction] = {} + theorem_terms = 0 + + def collect( + j: int, + indices: tuple[int, ...], + composition: tuple[int, ...], + ) -> None: + nonlocal theorem_terms + theorem_terms += 1 + outer: list[int] = [] + scalar = RationalInterval.point(_multinomial(order, composition)) + for stage_index, power in zip(indices, composition): + stage = stages[stage_index - 1] + scalar *= stage.coefficient**power + outer.extend([stage.fragment_index] * power) + scalar_upper = scalar.abs_upper() + for base_index in range(1, j): + base_stage = stages[base_index - 1] + key = tuple(outer) + (base_stage.fragment_index,) + weights[key] = weights.get(key, Fraction()) + ( + scalar_upper * base_stage.coefficient.abs_upper() + ) + + for j in range(2, center + 1): + indices = tuple(range(center, j - 1, -1)) + for composition in weak_compositions(order, len(indices)): + if composition[-1]: + collect(j, indices, composition) + for j in range(center + 1, count + 1): + indices = tuple(range(center + 1, j + 1)) + for composition in weak_compositions(order, len(indices)): + if composition[-1]: + collect(j, indices, composition) + + evaluator = SymplecticDyadicLocalDensityEvaluator(shared_coordinates=True) + total = sum( + ( + weight * evaluator.pauli_l1_density(key) + for key, weight in weights.items() + ), + Fraction(), + ) + cell = total / factorial(order + 1) + return FourthOrderPublishedTriangleCertificate( + center=center, + root_interval=root, + cell_density_upper=cell, + site_density_upper=cell / 4, + theorem_terms=theorem_terms, + expanded_commutator_keys=len(weights), + ) diff --git a/tracks/qcs/solutions/WangTheoPhys/issue128/src/trottercert/series.py b/tracks/qcs/solutions/WangTheoPhys/issue128/src/trottercert/series.py new file mode 100644 index 000000000..e6f67d72f --- /dev/null +++ b/tracks/qcs/solutions/WangTheoPhys/issue128/src/trottercert/series.py @@ -0,0 +1,107 @@ +from __future__ import annotations + +from dataclasses import dataclass +from fractions import Fraction +from typing import Iterable + +from .algebra import PauliSum, QComplex + + +@dataclass +class OperatorSeries: + """A truncated power series whose coefficients are exact operators.""" + + coefficients: list[PauliSum] + + @classmethod + def zero(cls, order: int) -> OperatorSeries: + return cls([PauliSum.zero() for _ in range(order + 1)]) + + @classmethod + def identity(cls, order: int) -> OperatorSeries: + result = cls.zero(order) + result.coefficients[0] = PauliSum.identity() + return result + + @property + def order(self) -> int: + return len(self.coefficients) - 1 + + def __add__(self, other: OperatorSeries) -> OperatorSeries: + if self.order != other.order: + raise ValueError("series truncation orders differ") + return OperatorSeries( + [left + right for left, right in zip(self.coefficients, other.coefficients)] + ) + + def __sub__(self, other: OperatorSeries) -> OperatorSeries: + return self + other.scale(-1) + + def scale(self, scalar: QComplex | int | Fraction) -> OperatorSeries: + return OperatorSeries([coefficient.scale(scalar) for coefficient in self.coefficients]) + + def __mul__(self, other: OperatorSeries) -> OperatorSeries: + if self.order != other.order: + raise ValueError("series truncation orders differ") + result = self.zero(self.order) + for degree in range(self.order + 1): + coefficient = PauliSum.zero() + for left_degree in range(degree + 1): + coefficient += ( + self.coefficients[left_degree] + * other.coefficients[degree - left_degree] + ) + result.coefficients[degree] = coefficient + return result + + def power(self, exponent: int) -> OperatorSeries: + if exponent < 0: + raise ValueError("negative series powers are unsupported") + result = self.identity(self.order) + base = self + for _ in range(exponent): + result = result * base + return result + + +def exponential_series( + generator: PauliSum, + coefficient: QComplex | int | Fraction, + order: int, +) -> OperatorSeries: + """Return exp(t * coefficient * generator) through ``t**order``.""" + + result = OperatorSeries.zero(order) + power = PauliSum.identity() + scalar = QComplex.coerce(coefficient) + scalar_power = QComplex(1) + factorial = 1 + for degree in range(order + 1): + if degree: + power = power * generator + scalar_power *= scalar + factorial *= degree + result.coefficients[degree] = power.scale(scalar_power / factorial) + return result + + +def product_series(stages: Iterable[OperatorSeries], order: int) -> OperatorSeries: + result = OperatorSeries.identity(order) + for stage in stages: + if stage.order != order: + raise ValueError("stage truncation order differs") + result = result * stage + return result + + +def logarithm_series(series: OperatorSeries) -> OperatorSeries: + if series.coefficients[0] != PauliSum.identity(): + raise ValueError("logarithm requires unit constant coefficient") + delta = series - OperatorSeries.identity(series.order) + result = OperatorSeries.zero(series.order) + power = OperatorSeries.identity(series.order) + for exponent in range(1, series.order + 1): + power = power * delta + sign = 1 if exponent % 2 else -1 + result = result + power.scale(Fraction(sign, exponent)) + return result diff --git a/tracks/qcs/solutions/WangTheoPhys/issue128/src/trottercert/su2clusters.py b/tracks/qcs/solutions/WangTheoPhys/issue128/src/trottercert/su2clusters.py new file mode 100644 index 000000000..1d59e0642 --- /dev/null +++ b/tracks/qcs/solutions/WangTheoPhys/issue128/src/trottercert/su2clusters.py @@ -0,0 +1,70 @@ +from __future__ import annotations + +from dataclasses import dataclass + +from .algebra import PauliSum +from .hamiltonian import heisenberg_bond +from .lattice import SquareLattice + + +@dataclass(frozen=True) +class LPathCluster: + color: int + center: int + left: int + up: int + + @property + def sites(self) -> frozenset[int]: + return frozenset((self.center, self.left, self.up)) + + @property + def bonds(self) -> frozenset[tuple[int, int]]: + return frozenset( + ( + tuple(sorted((self.center, self.left))), + tuple(sorted((self.center, self.up))), + ) + ) + + +def three_color_l_path_clusters( + lattice: SquareLattice, +) -> tuple[tuple[LPathCluster, ...], ...]: + """Square-lattice local-SU(2) decomposition from arXiv:2605.16016. + + A cluster centered at ``(x,y)`` contains the incoming horizontal and + vertical bonds and has color ``x-y mod 3``. + """ + + if lattice.length % 3: + raise ValueError("periodic three-color L-path decomposition requires 3 | L") + groups: list[list[LPathCluster]] = [[], [], []] + for y in range(lattice.length): + for x in range(lattice.length): + color = (x - y) % 3 + groups[color].append( + LPathCluster( + color=color, + center=lattice.site(x, y), + left=lattice.site(x - 1, y), + up=lattice.site(x, y - 1), + ) + ) + return tuple(tuple(group) for group in groups) + + +def l_path_cluster_hamiltonian(cluster: LPathCluster) -> PauliSum: + return heisenberg_bond(cluster.center, cluster.left) + heisenberg_bond( + cluster.center, cluster.up + ) + + +def three_l_path_fragments(lattice: SquareLattice) -> tuple[PauliSum, ...]: + return tuple( + sum( + (l_path_cluster_hamiltonian(cluster) for cluster in group), + PauliSum.zero(), + ) + for group in three_color_l_path_clusters(lattice) + ) diff --git a/tracks/qcs/solutions/WangTheoPhys/issue128/src/trottercert/support_groups.py b/tracks/qcs/solutions/WangTheoPhys/issue128/src/trottercert/support_groups.py new file mode 100644 index 000000000..c35b1650c --- /dev/null +++ b/tracks/qcs/solutions/WangTheoPhys/issue128/src/trottercert/support_groups.py @@ -0,0 +1,292 @@ +from __future__ import annotations + +from collections.abc import Mapping, Sequence +from fractions import Fraction +import gzip +import json +from math import lcm +from typing import Any + +from .anticommuting import ( + AnticommutingPartitionCertificate, + certify_anticommuting_partition, + symplectic_anticommutes, +) +from .intervals import RationalInterval +from .local_commutators import SymplecticPauli + + +def _support(pauli: SymplecticPauli) -> int: + return pauli[0] | pauli[1] + + +def discover_support_partition( + coefficients: Mapping[SymplecticPauli, RationalInterval], + *, + max_group_size: int = 10, + discovery_scores: Mapping[SymplecticPauli, Fraction] | None = None, +) -> tuple[tuple[SymplecticPauli, ...], ...]: + """Discover anticommuting groups without mixing Pauli supports.""" + + if max_group_size < 1: + raise ValueError("maximum group size must be positive") + if discovery_scores is not None and set(discovery_scores) != set( + coefficients + ): + raise ValueError("discovery scores must cover the coefficient map") + buckets: dict[int, dict[SymplecticPauli, RationalInterval]] = {} + for pauli, coefficient in coefficients.items(): + buckets.setdefault(_support(pauli), {})[pauli] = coefficient + groups: list[tuple[SymplecticPauli, ...]] = [] + for support in sorted(buckets): + bucket = buckets[support] + ordered = tuple( + sorted( + bucket, + key=lambda pauli: ( + -( + abs(discovery_scores[pauli]) + if discovery_scores is not None + else abs(bucket[pauli].midpoint()) + ), + pauli, + ), + ) + ) + used: set[SymplecticPauli] = set() + for position, pauli in enumerate(ordered): + if pauli in used: + continue + group = [pauli] + used.add(pauli) + for candidate in ordered[position + 1 :]: + if candidate in used: + continue + if all( + symplectic_anticommutes(candidate, member) + for member in group + ): + group.append(candidate) + used.add(candidate) + if len(group) == max_group_size: + break + groups.append(tuple(group)) + return tuple(groups) + + +def certify_support_partition( + coefficients: Mapping[SymplecticPauli, RationalInterval], + groups: Sequence[Sequence[SymplecticPauli]], + *, + sqrt_decimal_places: int = 30, +) -> AnticommutingPartitionCertificate: + """Verify coverage, equal supports, and pairwise anticommutation.""" + + flattened = tuple(pauli for group in groups for pauli in group) + if len(flattened) != len(set(flattened)): + raise ValueError("partition coverage contains duplicate terms") + if set(flattened) != set(coefficients): + raise ValueError("partition coverage differs from coefficient map") + for group in groups: + if not group: + raise ValueError("anticommuting group must be nonempty") + supports = {_support(pauli) for pauli in group} + if len(supports) != 1: + raise ValueError("group members must have the same support") + return certify_anticommuting_partition( + coefficients, + groups, + sqrt_decimal_places=sqrt_decimal_places, + ) + + +def _pair(value: Fraction) -> list[int]: + return [value.numerator, value.denominator] + + +def _canonical_json_bytes(payload: object) -> bytes: + return ( + json.dumps(payload, sort_keys=True, separators=(",", ":")) + "\n" + ).encode() + + +def canonical_gzip_bytes(payload: object) -> bytes: + """Return deterministic gzip bytes for a canonical JSON payload.""" + + compressed = bytearray( + gzip.compress(_canonical_json_bytes(payload), compresslevel=9, mtime=0) + ) + # CPython has changed how ``gzip.compress(..., mtime=0)`` chooses the + # RFC 1952 OS header byte. Python 3.14 emits 0xff here, whereas the + # frozen schema-v3 sidecar was produced with 0x13 on macOS. The byte has + # no effect on decompression, but it is part of the proof artifact's + # SHA-256. Pin the historical canonical value explicitly so rebuilding + # remains byte-for-byte stable across Python and platform upgrades. + compressed[9] = 0x13 + return bytes(compressed) + + +def build_d5_payload( + coefficients: Mapping[SymplecticPauli, RationalInterval], + groups: Sequence[Sequence[SymplecticPauli]] | None = None, +) -> dict[str, object]: + selected_groups = ( + discover_support_partition(coefficients, max_group_size=10) + if groups is None + else tuple(tuple(group) for group in groups) + ) + certificate = certify_support_partition(coefficients, selected_groups) + coefficient_denominator = 1 + for interval in certificate.coefficients: + coefficient_denominator = lcm( + coefficient_denominator, + interval.lower.denominator, + interval.upper.denominator, + ) + sqrt_denominator = 1 + for group in certificate.groups: + sqrt_denominator = lcm(sqrt_denominator, group.bound.denominator) + return { + "schema_version": 1, + "kind": "issue128_exact_d5_support_partition", + "coefficient_denominator": coefficient_denominator, + "terms": [ + [ + pauli[0], + pauli[1], + interval.lower.numerator + * (coefficient_denominator // interval.lower.denominator), + interval.upper.numerator + * (coefficient_denominator // interval.upper.denominator), + ] + for pauli, interval in zip( + certificate.paulis, + certificate.coefficients, + ) + ], + "sqrt_denominator": sqrt_denominator, + "groups": [ + [ + list(group.term_indices), + group.bound.numerator + * (sqrt_denominator // group.bound.denominator), + ] + for group in certificate.groups + ], + "term_count": len(certificate.paulis), + "group_count": len(certificate.groups), + "cell_bound": _pair(certificate.bound), + "site_bound": _pair(certificate.bound / 4), + } + + +def _integer(value: object, *, field: str) -> int: + if isinstance(value, bool) or not isinstance(value, int): + raise ValueError(f"{field} must be an integer") + return value + + +def _fraction_pair(value: object, *, field: str) -> Fraction: + if not isinstance(value, list) or len(value) != 2: + raise ValueError(f"{field} must be a rational pair") + numerator = _integer(value[0], field=f"{field} numerator") + denominator = _integer(value[1], field=f"{field} denominator") + if denominator <= 0: + raise ValueError(f"{field} denominator must be positive") + return Fraction(numerator, denominator) + + +def verify_d5_payload( + payload: Mapping[str, object], +) -> AnticommutingPartitionCertificate: + if payload.get("schema_version") != 1: + raise ValueError("unsupported D5 sidecar schema") + if payload.get("kind") != "issue128_exact_d5_support_partition": + raise ValueError("unexpected D5 sidecar kind") + coefficient_denominator = _integer( + payload.get("coefficient_denominator"), + field="coefficient denominator", + ) + sqrt_denominator = _integer( + payload.get("sqrt_denominator"), + field="sqrt denominator", + ) + if coefficient_denominator <= 0 or sqrt_denominator <= 0: + raise ValueError("D5 sidecar denominators must be positive") + raw_terms = payload.get("terms") + raw_groups = payload.get("groups") + if not isinstance(raw_terms, list) or not isinstance(raw_groups, list): + raise ValueError("D5 sidecar terms and groups must be lists") + + paulis: list[SymplecticPauli] = [] + coefficients: dict[SymplecticPauli, RationalInterval] = {} + for raw_term in raw_terms: + if not isinstance(raw_term, list) or len(raw_term) != 4: + raise ValueError("D5 sidecar term is malformed") + pauli = ( + _integer(raw_term[0], field="term x mask"), + _integer(raw_term[1], field="term z mask"), + ) + if pauli in coefficients: + raise ValueError("D5 sidecar contains duplicate terms") + lower = _integer(raw_term[2], field="term lower numerator") + upper = _integer(raw_term[3], field="term upper numerator") + coefficients[pauli] = RationalInterval( + Fraction(lower, coefficient_denominator), + Fraction(upper, coefficient_denominator), + ) + paulis.append(pauli) + if paulis != sorted(paulis): + raise ValueError("D5 sidecar terms are not canonically sorted") + + groups: list[tuple[SymplecticPauli, ...]] = [] + encoded_bounds: list[Fraction] = [] + for raw_group in raw_groups: + if not isinstance(raw_group, list) or len(raw_group) != 2: + raise ValueError("D5 sidecar group is malformed") + indices = raw_group[0] + if not isinstance(indices, list): + raise ValueError("D5 sidecar group indices must be a list") + try: + group = tuple( + paulis[_integer(index, field="group term index")] + for index in indices + ) + except IndexError as error: + raise ValueError("D5 sidecar group index is out of range") from error + groups.append(group) + encoded_bounds.append( + Fraction( + _integer(raw_group[1], field="group bound numerator"), + sqrt_denominator, + ) + ) + + certificate = certify_support_partition(coefficients, groups) + if encoded_bounds != [group.bound for group in certificate.groups]: + raise ValueError("D5 sidecar group bound mismatch") + if _integer(payload.get("term_count"), field="term count") != len(paulis): + raise ValueError("D5 sidecar term count mismatch") + if _integer(payload.get("group_count"), field="group count") != len(groups): + raise ValueError("D5 sidecar group count mismatch") + if _fraction_pair(payload.get("cell_bound"), field="cell bound") != ( + certificate.bound + ): + raise ValueError("D5 sidecar cell bound mismatch") + if _fraction_pair(payload.get("site_bound"), field="site bound") != ( + certificate.bound / 4 + ): + raise ValueError("D5 sidecar site bound mismatch") + return certificate + + +def decode_d5_gzip(raw: bytes) -> dict[str, object]: + try: + payload: Any = json.loads(gzip.decompress(raw)) + except (OSError, json.JSONDecodeError) as error: + raise ValueError("D5 sidecar is not canonical gzip JSON") from error + if not isinstance(payload, dict): + raise ValueError("D5 sidecar root must be an object") + if canonical_gzip_bytes(payload) != raw: + raise ValueError("D5 sidecar bytes are not canonical") + return payload diff --git a/tracks/qcs/solutions/WangTheoPhys/issue128/src/trottercert/verify.py b/tracks/qcs/solutions/WangTheoPhys/issue128/src/trottercert/verify.py new file mode 100644 index 000000000..ac1c125a8 --- /dev/null +++ b/tracks/qcs/solutions/WangTheoPhys/issue128/src/trottercert/verify.py @@ -0,0 +1,530 @@ +from __future__ import annotations + +import hashlib +import json +from dataclasses import dataclass +from fractions import Fraction +from pathlib import Path + +from .anticommuting import certify_anticommuting_partition +from .baseline import pauli_l1_second_order_constant +from .hamiltonian import four_matching_fragments +from .intervals import RationalInterval, cube_root_four_interval +from .lattice import SquareLattice +from .local_commutators import SymplecticPauli +from .resources import ( + fourth_order_four_matching_resources, + four_matching_resources, + required_steps, + three_l_path_resources, +) +from .su2clusters import three_l_path_fragments + + +EXPECTED_NORMALIZATION = "(XX+YY+ZZ)/4" + + +@dataclass(frozen=True) +class D4SidecarVerification: + site_bound: Fraction + coefficients: dict[SymplecticPauli, RationalInterval] + term_count: int + group_count: int + max_group_size: int + + +def _fraction(pair: list[int]) -> Fraction: + if len(pair) != 2: + raise ValueError("fraction must be [numerator, denominator]") + return Fraction(pair[0], pair[1]) + + +def _verify_d4_sidecar( + certificate_path: Path, + candidate: dict[str, object], +) -> D4SidecarVerification: + metadata = candidate["d4_certificate"] + root = certificate_path.resolve().parent + sidecar_path = (root / str(metadata["path"])).resolve() + if sidecar_path.parent != root: + raise ValueError("D4 sidecar path escapes certificate directory") + raw = sidecar_path.read_bytes() + if hashlib.sha256(raw).hexdigest() != str(metadata["sha256"]): + raise ValueError("D4 sidecar digest mismatch") + + payload = json.loads(raw) + if int(payload["schema_version"]) != 1: + raise ValueError("unsupported D4 sidecar schema") + coefficient_denominator = int(payload["coefficient_denominator"]) + sqrt_denominator = int(payload["sqrt_denominator"]) + if coefficient_denominator < 1 or sqrt_denominator < 1: + raise ValueError("D4 sidecar denominators must be positive") + + rows = payload["terms"] + paulis = tuple( + (int(row[0]), int(row[1])) + for row in rows + ) + if len(paulis) != len(set(paulis)): + raise ValueError("D4 sidecar coefficient terms are duplicated") + if paulis != tuple(sorted(paulis)): + raise ValueError("D4 sidecar coefficient terms are not canonical") + coefficients = { + pauli: RationalInterval( + Fraction(int(row[2]), coefficient_denominator), + Fraction(int(row[3]), coefficient_denominator), + ) + for pauli, row in zip(paulis, rows) + } + + submitted_groups = payload["groups"] + groups: list[tuple[SymplecticPauli, ...]] = [] + for row in submitted_groups: + indices = tuple(int(index) for index in row[0]) + if any(index < 0 or index >= len(paulis) for index in indices): + raise ValueError("D4 partition coverage index is out of range") + groups.append(tuple(paulis[index] for index in indices)) + regenerated = certify_anticommuting_partition( + coefficients, + tuple(groups), + ) + submitted_bounds = tuple( + Fraction(int(row[1]), sqrt_denominator) + for row in submitted_groups + ) + regenerated_bounds = tuple( + group.bound for group in regenerated.groups + ) + if submitted_bounds != regenerated_bounds: + raise ValueError("D4 group bound mismatch") + + max_group_size = max((len(group) for group in groups), default=0) + if max_group_size > int(metadata["max_group_size"]): + raise ValueError("D4 maximum group size mismatch") + cell_bound = _fraction(payload["cell_bound"]) + if cell_bound != regenerated.bound: + raise ValueError("D4 cell bound mismatch") + if cell_bound != _fraction(metadata["cell_norm_upper"]): + raise ValueError("D4 main-certificate bound mismatch") + return D4SidecarVerification( + site_bound=cell_bound / 4, + coefficients=coefficients, + term_count=len(paulis), + group_count=len(groups), + max_group_size=max_group_size, + ) + + +def _verify_v1(data: dict[str, object]) -> dict[str, object]: + benchmark = data["benchmark"] + if benchmark["normalization"] != EXPECTED_NORMALIZATION: + raise ValueError("Hamiltonian normalization mismatch") + length = int(benchmark["length"]) + if length % 6: + raise ValueError("benchmark length must be divisible by six") + tolerance = _fraction(benchmark["tolerance"]) + reference_length = int(data["proof"]["reference_length"]) + if reference_length != 6: + raise ValueError("the verifier currently pins reference length L=6") + + lattice = SquareLattice(reference_length) + baseline_ref = pauli_l1_second_order_constant( + four_matching_fragments(lattice) + ) + candidate_ref = pauli_l1_second_order_constant( + three_l_path_fragments(lattice) + ) + baseline_density = baseline_ref / lattice.n_sites + candidate_density = candidate_ref / lattice.n_sites + if baseline_density != _fraction(data["proof"]["baseline_density"]): + raise ValueError("baseline density summary is inconsistent") + if candidate_density != _fraction(data["proof"]["candidate_density"]): + raise ValueError("candidate density summary is inconsistent") + + n_sites = length * length + baseline_constant = baseline_density * n_sites + candidate_constant = candidate_density * n_sites + baseline_steps = required_steps(baseline_constant, tolerance, 2) + candidate_steps = required_steps(candidate_constant, tolerance, 2) + baseline = four_matching_resources(n_sites, baseline_steps) + candidate = three_l_path_resources(n_sites, candidate_steps) + + claimed = data["claimed_resources"] + recomputed = { + "baseline_steps": baseline.steps, + "candidate_steps": candidate.steps, + "baseline_local_propagators": baseline.local_propagators, + "candidate_local_propagators": candidate.local_propagators, + "baseline_cnot_upper": baseline.cnot_upper, + "candidate_cnot_upper": candidate.cnot_upper, + } + if claimed != recomputed: + raise ValueError("claimed resources do not match recomputation") + local_ratio = Fraction( + candidate.local_propagators, + baseline.local_propagators, + ) + return { + "valid": True, + "baseline_density": str(baseline_density), + "candidate_density": str(candidate_density), + "local_propagator_ratio": str(local_ratio), + "local_propagator_improvement": float(1 / local_ratio), + "compiled_cnot_improvement": ( + baseline.cnot_upper / candidate.cnot_upper + ), + "twofold_local_target_met": local_ratio <= Fraction(1, 2), + "twofold_compiled_cnot_target_met": ( + candidate.cnot_upper * 2 <= baseline.cnot_upper + ), + } + + +def _verify_v2( + data: dict[str, object], + *, + deep: bool, +) -> dict[str, object]: + benchmark = data["benchmark"] + if benchmark["normalization"] != EXPECTED_NORMALIZATION: + raise ValueError("Hamiltonian normalization mismatch") + length = int(benchmark["length"]) + if length % 2: + raise ValueError("four-matching benchmark requires even length") + n_sites = length * length + time = _fraction(benchmark["time"]) + tolerance = _fraction(benchmark["tolerance"]) + if time != 1: + raise ValueError("schema v2 currently pins T=1") + + proof = data["proof"] + if proof["formula"] != "five_copy_suzuki_fourth_order": + raise ValueError("unexpected candidate formula") + if int(proof["stage_count"]) != 31 or int(proof["center"]) != 17: + raise ValueError("candidate formula structure mismatch") + decimal_digits = int(proof["coefficient_interval_decimal_digits"]) + root = cube_root_four_interval(decimal_digits) + claimed_root = proof["cube_root_four_interval"] + if ( + root.lower != _fraction(claimed_root["lower"]) + or root.upper != _fraction(claimed_root["upper"]) + or not (root.lower**3 <= 4 <= root.upper**3) + ): + raise ValueError("cube-root enclosure is invalid") + + baseline_density = _fraction(proof["baseline_site_density"]) + if baseline_density <= 0 or baseline_density > Fraction(21, 8): + raise ValueError("strengthened baseline density is outside the pinned range") + candidate_density = _fraction(proof["candidate_site_density_upper"]) + if candidate_density <= 0: + raise ValueError("candidate density must be positive") + + deep_verified = False + if deep: + from .baseline import strang_commutator_operators + from .clusters import phase_partitioned_collatz_certificate + from .rigorous_fourth import fourth_order_rational_pair_certificate + + reference_length = int(proof["baseline_reference_length"]) + if reference_length != 6: + raise ValueError("deep baseline currently pins L=6") + reference_lattice = SquareLattice(reference_length) + baseline_total = Fraction() + component_bounds: list[list[list[int]]] = [] + for block in strang_commutator_operators( + four_matching_fragments(reference_lattice) + ): + repeated_fragment = phase_partitioned_collatz_certificate( + block.repeated_fragment, + reference_lattice, + iterations=35, + ).global_bound + repeated_tail = phase_partitioned_collatz_certificate( + block.repeated_tail, + reference_lattice, + iterations=35, + ).global_bound + baseline_total += repeated_fragment / 24 + repeated_tail / 12 + component_bounds.append( + [ + [repeated_fragment.numerator, repeated_fragment.denominator], + [repeated_tail.numerator, repeated_tail.denominator], + ] + ) + regenerated_baseline_density = ( + baseline_total / reference_lattice.n_sites + ) + if regenerated_baseline_density != baseline_density: + raise ValueError("deep regenerated baseline differs from certificate") + if component_bounds != proof["baseline_component_bounds"]: + raise ValueError("deep baseline component bounds differ") + + regenerated = fourth_order_rational_pair_certificate( + center=17, + decimal_digits=decimal_digits, + ) + if regenerated.site_density_upper != candidate_density: + raise ValueError("deep regenerated density differs from certificate") + statistics = proof["statistics"] + if ( + regenerated.theorem_terms != int(statistics["theorem_terms"]) + or regenerated.paired_terms != int(statistics["paired_terms"]) + or regenerated.singleton_terms != int(statistics["singleton_terms"]) + ): + raise ValueError("deep proof statistics differ from certificate") + deep_verified = True + + baseline_steps = required_steps( + baseline_density * n_sites, + tolerance, + 2, + time, + ) + candidate_steps = required_steps( + candidate_density * n_sites, + tolerance, + 4, + time, + ) + baseline = four_matching_resources(n_sites, baseline_steps) + candidate = fourth_order_four_matching_resources( + n_sites, + candidate_steps, + stage_count=31, + ) + claimed = data["claimed_resources"] + recomputed = { + "baseline_steps": baseline.steps, + "candidate_steps": candidate.steps, + "baseline_group_exponentials": baseline.group_exponentials, + "candidate_group_exponentials": candidate.group_exponentials, + "baseline_bond_propagators": baseline.local_propagators, + "candidate_bond_propagators": candidate.local_propagators, + "baseline_cnot_upper": baseline.cnot_upper, + "candidate_cnot_upper": candidate.cnot_upper, + } + if claimed != recomputed: + raise ValueError("claimed resources do not match recomputation") + + ratio = Fraction(candidate.cnot_upper, baseline.cnot_upper) + twofold = candidate.cnot_upper * 2 <= baseline.cnot_upper + if bool(data["claims"]["compiled_cnot_improvement_exceeds_two"]) != twofold: + raise ValueError("twofold claim is inconsistent") + return { + "valid": True, + "verification_level": "deep" if deep_verified else "fast", + "deep_proof_regenerated": deep_verified, + "baseline_site_density": str(baseline_density), + "candidate_site_density_upper": str(candidate_density), + "baseline_steps": baseline.steps, + "candidate_steps": candidate.steps, + "compiled_cnot_ratio": str(ratio), + "compiled_cnot_improvement": float(1 / ratio), + "twofold_compiled_cnot_target_met": twofold, + } + + +def _verify_v3( + data: dict[str, object], + *, + deep: bool, + certificate_path: Path, +) -> dict[str, object]: + benchmark = data["benchmark"] + if benchmark["normalization"] != EXPECTED_NORMALIZATION: + raise ValueError("Hamiltonian normalization mismatch") + length = int(benchmark["length"]) + if length % 2: + raise ValueError("four-matching benchmark requires even length") + n_sites = length * length + if _fraction(benchmark["time"]) != 1: + raise ValueError("schema v3 pins T=1") + tolerance = _fraction(benchmark["tolerance"]) + + published = data["published_baseline"] + if ( + published["formula"] != "five_copy_suzuki_fourth_order" + or int(published["formula_order"]) != 4 + or int(published["stage_count"]) != 31 + or int(published["theorem_center"]) != 20 + ): + raise ValueError("published baseline structure mismatch") + published_density = _fraction(published["site_density_upper"]) + published_steps = required_steps( + published_density * n_sites, + tolerance, + 4, + ) + if published_steps != int(published["steps"]): + raise ValueError("published baseline step count mismatch") + published_groups = 30 * published_steps + 1 + if published_groups != int(published["group_exponentials"]): + raise ValueError("published baseline group count mismatch") + + candidate = data["candidate"] + if ( + candidate["formula"] != "five_copy_suzuki_fourth_order" + or candidate["proof_method"] + != ( + "local_log_E5_grouped_D4_plus_E7_majorant" + "_plus_exact_generator_tail" + ) + ): + raise ValueError("candidate structure mismatch") + d4_verification = _verify_d4_sidecar( + certificate_path, + candidate, + ) + d4_metadata = candidate["d4_certificate"] + if d4_verification.term_count != int(d4_metadata["term_count"]): + raise ValueError("D4 term count mismatch") + if d4_verification.group_count != int(d4_metadata["group_count"]): + raise ValueError("D4 group count mismatch") + if ( + d4_verification.max_group_size + > int(d4_metadata["max_group_size"]) + ): + raise ValueError("D4 maximum group size mismatch") + candidate_steps = int(candidate["steps"]) + candidate_error = _fraction(candidate["global_error_upper"]) + previous_error = _fraction(candidate["previous_step_error_upper"]) + if candidate_error > tolerance: + raise ValueError("candidate does not meet the requested tolerance") + if previous_error <= tolerance: + raise ValueError("candidate step count is not minimal for this certificate") + candidate_groups = 30 * candidate_steps + 1 + if candidate_groups != int(candidate["group_exponentials"]): + raise ValueError("candidate group count mismatch") + contributions = candidate["contributions"] + contribution_sum = sum( + (_fraction(contributions[key]) for key in ("degree4", "degree5", "degree6", "degree7", "tail")), + Fraction(), + ) + if contribution_sum != candidate_error: + raise ValueError("candidate contribution sum mismatch") + expected_degree_four = ( + Fraction(n_sites) + * d4_verification.site_bound + / (5 * candidate_steps**4) + ) + if _fraction(contributions["degree4"]) != expected_degree_four: + raise ValueError("candidate grouped D4 contribution mismatch") + + claimed = data["claimed_resources"] + baseline_bonds = published_groups * n_sites // 2 + candidate_bonds = candidate_groups * n_sites // 2 + recomputed = { + "published_steps": published_steps, + "candidate_steps": candidate_steps, + "published_group_exponentials": published_groups, + "candidate_group_exponentials": candidate_groups, + "published_bond_propagators": baseline_bonds, + "candidate_bond_propagators": candidate_bonds, + "published_cnot_upper": 3 * baseline_bonds, + "candidate_cnot_upper": 3 * candidate_bonds, + } + if claimed != recomputed: + raise ValueError("schema v3 resource summary mismatch") + + deep_verified = False + if deep: + from .refined_error import ( + build_refined_fourth_order_constants, + certified_d4_cell_coefficients, + evaluate_refined_fourth_order_bound, + ) + from .rigorous_fourth import ( + fourth_order_published_triangle_certificate, + ) + + published_rebuilt = fourth_order_published_triangle_certificate( + center=20, + decimal_digits=int(published["coefficient_interval_decimal_digits"]), + ) + if published_rebuilt.site_density_upper != published_density: + raise ValueError("deep published baseline regeneration mismatch") + + constants = build_refined_fourth_order_constants( + decimal_digits=int(candidate["coefficient_interval_decimal_digits"]), + quantization_digits=int(candidate["e5_quantization_digits"]), + ) + if constants.e5_site_l1 != _fraction(candidate["e5_site_l1_upper"]): + raise ValueError("deep E5 regeneration mismatch") + if constants.e7_site_majorant != _fraction(candidate["e7_site_majorant"]): + raise ValueError("deep E7 regeneration mismatch") + regenerated_d4 = certified_d4_cell_coefficients( + constants.stages, + quantization_digits=int(candidate["e5_quantization_digits"]), + ) + if regenerated_d4 != d4_verification.coefficients: + raise ValueError("deep D4 coefficient regeneration mismatch") + rebuilt = evaluate_refined_fourth_order_bound( + constants, + n_sites, + candidate_steps, + d4_site_override=d4_verification.site_bound, + ) + rebuilt_previous = evaluate_refined_fourth_order_bound( + constants, + n_sites, + candidate_steps - 1, + d4_site_override=d4_verification.site_bound, + ) + if rebuilt.global_error_bound != candidate_error: + raise ValueError("deep candidate bound regeneration mismatch") + if rebuilt_previous.global_error_bound != previous_error: + raise ValueError("deep candidate minimality regeneration mismatch") + deep_verified = True + + ratio = Fraction(published_groups, candidate_groups) + claimed_ratio = _fraction(data["claims"]["exact_improvement_ratio"]) + if ratio != claimed_ratio: + raise ValueError("claimed improvement ratio mismatch") + if bool(data["claims"]["global_twofold_target_met"]) != (ratio >= 2): + raise ValueError("global twofold claim mismatch") + if bool(data["claims"]["global_fourfold_target_met"]) != (ratio >= 4): + raise ValueError("global fourfold claim mismatch") + return { + "valid": True, + "verification_level": "deep" if deep_verified else "fast", + "deep_proof_regenerated": deep_verified, + "published_steps": published_steps, + "candidate_steps": candidate_steps, + "published_group_exponentials": published_groups, + "candidate_group_exponentials": candidate_groups, + "candidate_error_upper": str(candidate_error), + "previous_step_error_upper": str(previous_error), + "d4_cell_norm_upper": str( + 4 * d4_verification.site_bound + ), + "d4_term_count": d4_verification.term_count, + "d4_group_count": d4_verification.group_count, + "d4_max_group_size": d4_verification.max_group_size, + "exact_improvement_ratio": str(ratio), + "improvement": float(ratio), + "global_twofold_target_met": ratio >= 2, + "global_fourfold_target_met": ratio >= 4, + } + + +def verify_certificate( + path: str | Path, + *, + deep: bool = False, +) -> dict[str, object]: + certificate_path = Path(path) + data = json.loads(certificate_path.read_text()) + schema = data.get("schema_version") + if schema == 1: + if deep: + raise ValueError("deep verification is available only for schema v2") + return _verify_v1(data) + if schema == 2: + return _verify_v2(data, deep=deep) + if schema == 3: + return _verify_v3( + data, + deep=deep, + certificate_path=certificate_path, + ) + raise ValueError("unsupported certificate schema") diff --git a/tracks/qcs/solutions/WangTheoPhys/issue128/tests/test_algebra.py b/tracks/qcs/solutions/WangTheoPhys/issue128/tests/test_algebra.py new file mode 100644 index 000000000..48f0d2dc7 --- /dev/null +++ b/tracks/qcs/solutions/WangTheoPhys/issue128/tests/test_algebra.py @@ -0,0 +1,48 @@ +from fractions import Fraction + +import numpy as np + +from trottercert.algebra import ( + PauliString, + PauliSum, + QComplex, + commutator, + to_dense, + pauli_strings_commute, +) + + +def term(site: int, op: str) -> PauliSum: + return PauliSum.term(PauliString({site: op})) + + +def test_single_qubit_pauli_phase_table() -> None: + x, y, z = term(0, "X"), term(0, "Y"), term(0, "Z") + assert x * y == z.scale(QComplex(0, 1)) + assert y * x == z.scale(QComplex(0, -1)) + assert commutator(x, y) == z.scale(QComplex(0, 2)) + assert not pauli_strings_commute(PauliString({0: "X"}), PauliString({0: "Y"})) + assert pauli_strings_commute(PauliString({0: "X"}), PauliString({1: "Y"})) + + +def test_multiqubit_multiplication_and_merging() -> None: + xx = PauliSum.term(PauliString({0: "X", 2: "X"}), Fraction(1, 4)) + yy = PauliSum.term(PauliString({0: "Y", 2: "Y"}), Fraction(1, 4)) + product = xx * yy + assert product == PauliSum.term(PauliString({0: "Z", 2: "Z"}), Fraction(-1, 16)) + assert xx - xx == PauliSum.zero() + + +def test_dense_agrees_with_exact_algebra() -> None: + left = term(0, "X") + term(1, "Z").scale(Fraction(2, 3)) + right = term(0, "Y") * term(1, "X") + expected = to_dense(left, 2) @ to_dense(right, 2) + assert np.allclose(to_dense(left * right, 2), expected) + + +def test_dagger_and_l1() -> None: + op = PauliSum.term(PauliString({0: "X"}), QComplex(Fraction(1, 2), Fraction(1, 3))) + assert op.dagger().terms[PauliString({0: "X"})] == QComplex( + Fraction(1, 2), Fraction(-1, 3) + ) + assert np.isclose(op.pauli_l1(), np.hypot(0.5, 1 / 3)) diff --git a/tracks/qcs/solutions/WangTheoPhys/issue128/tests/test_anticommuting.py b/tracks/qcs/solutions/WangTheoPhys/issue128/tests/test_anticommuting.py new file mode 100644 index 000000000..0417937ae --- /dev/null +++ b/tracks/qcs/solutions/WangTheoPhys/issue128/tests/test_anticommuting.py @@ -0,0 +1,90 @@ +from fractions import Fraction + +import pytest + +from trottercert.anticommuting import ( + certify_anticommuting_partition, + discover_anticommuting_partition, + sqrt_fraction_upper, + symplectic_anticommutes, +) +from trottercert.intervals import RationalInterval + + +def test_symplectic_anticommutation() -> None: + x = (1, 0) + y = (1, 1) + z = (0, 1) + assert symplectic_anticommutes(x, y) + assert symplectic_anticommutes(y, z) + assert symplectic_anticommutes(z, x) + assert not symplectic_anticommutes(x, x) + + +def test_rational_square_root_is_outward() -> None: + bound = sqrt_fraction_upper(Fraction(2), decimal_places=12) + assert bound * bound >= 2 + assert bound - Fraction(1414213562373, 10**12) <= Fraction(1, 10**12) + + +def test_exact_group_bound_uses_euclidean_norm() -> None: + coefficients = { + (1, 0): RationalInterval.point(1), + (1, 1): RationalInterval.point(2), + (0, 1): RationalInterval.point(2), + } + certificate = certify_anticommuting_partition( + coefficients, + (((1, 0), (1, 1), (0, 1)),), + ) + assert certificate.bound == 3 + assert certificate.groups[0].term_indices == (1, 2, 0) + + +def test_corrupt_commuting_group_is_rejected() -> None: + coefficients = { + (1, 0): RationalInterval.point(1), + (2, 0): RationalInterval.point(1), + } + with pytest.raises(ValueError, match="do not anticommute"): + certify_anticommuting_partition( + coefficients, + (((1, 0), (2, 0)),), + ) + + +def test_partition_coverage_rejects_duplicates_and_omissions() -> None: + coefficients = { + (1, 0): RationalInterval.point(1), + (1, 1): RationalInterval.point(1), + } + with pytest.raises(ValueError, match="coverage contains duplicate"): + certify_anticommuting_partition( + coefficients, + (((1, 0),), ((1, 0), (1, 1))), + ) + with pytest.raises(ValueError, match="coverage differs"): + certify_anticommuting_partition( + coefficients, + (((1, 0),),), + ) + + +def test_discovery_is_deterministic_and_covers_terms() -> None: + coefficients = { + (1, 0): RationalInterval.point(3), + (1, 1): RationalInterval.point(2), + (0, 1): RationalInterval.point(1), + } + first = discover_anticommuting_partition( + coefficients, + max_group_size=3, + ) + second = discover_anticommuting_partition( + coefficients, + max_group_size=3, + ) + assert first == second + certificate = certify_anticommuting_partition(coefficients, first) + assert set(certificate.paulis) == set(coefficients) + assert certificate.bound == sqrt_fraction_upper(Fraction(14)) diff --git a/tracks/qcs/solutions/WangTheoPhys/issue128/tests/test_baseline.py b/tracks/qcs/solutions/WangTheoPhys/issue128/tests/test_baseline.py new file mode 100644 index 000000000..e59002d1b --- /dev/null +++ b/tracks/qcs/solutions/WangTheoPhys/issue128/tests/test_baseline.py @@ -0,0 +1,30 @@ +from fractions import Fraction + +from trottercert.baseline import ( + pauli_l1_second_order_constant, + strang_commutator_operators, +) +from trottercert.hamiltonian import four_matching_fragments +from trottercert.lattice import SquareLattice + + +def test_baseline_blocks_are_hermitian() -> None: + fragments = four_matching_fragments(SquareLattice(4)) + blocks = strang_commutator_operators(fragments) + assert len(blocks) == 3 + assert all(block.repeated_fragment.is_hermitian() for block in blocks) + assert all(block.repeated_tail.is_hermitian() for block in blocks) + + +def test_pauli_l1_baseline_l4() -> None: + fragments = four_matching_fragments(SquareLattice(4)) + constant = pauli_l1_second_order_constant(fragments) + assert constant == Fraction(21, 8) * 16 + + +def test_symbolic_aggregate_coefficients_are_real_l4() -> None: + fragments = four_matching_fragments(SquareLattice(4)) + for block in strang_commutator_operators(fragments): + for operator in (block.repeated_fragment, block.repeated_tail): + assert operator.is_hermitian() + assert all(coefficient.imag == 0 for coefficient in operator.terms.values()) diff --git a/tracks/qcs/solutions/WangTheoPhys/issue128/tests/test_clusters.py b/tracks/qcs/solutions/WangTheoPhys/issue128/tests/test_clusters.py new file mode 100644 index 000000000..c8442a2fb --- /dev/null +++ b/tracks/qcs/solutions/WangTheoPhys/issue128/tests/test_clusters.py @@ -0,0 +1,90 @@ +from fractions import Fraction + +import numpy as np + +from trottercert.algebra import PauliString, PauliSum, to_dense +from trottercert.clusters import ( + _sqrt_fraction_upper, + build_uniform_cluster_certificate, + computational_row_sum_bound, + computational_row_taxicab_bound, + build_partitioned_operator_certificate, + build_full_patch_operator_certificate, + collatz_taxicab_certificate, + greedy_anticommuting_certificate, + paulis_anticommute, + phase_partitioned_collatz_certificate, +) +from trottercert.lattice import SquareLattice +from trottercert.orbits import canonical_translation, translation_orbits + + +def test_rational_sqrt_upper_is_outward() -> None: + bound = _sqrt_fraction_upper(Fraction(2), decimal_places=12) + assert bound * bound >= 2 + assert float(bound) - np.sqrt(2) < 1e-11 + + +def test_anticommuting_group_norm_bound() -> None: + x = PauliString({0: "X"}) + y = PauliString({0: "Y"}) + z = PauliString({0: "Z"}) + assert paulis_anticommute(x, y) + operator = PauliSum({x: 1, y: 2, z: 2}) + certificate = greedy_anticommuting_certificate(operator) + assert len(certificate.groups) == 1 + assert certificate.bound * certificate.bound >= 9 + assert float(certificate.bound) < 3 + 1e-12 + + +def test_row_sum_bound_upper_bounds_spectral_norm() -> None: + operator = ( + PauliSum.term(PauliString({0: "X"}), Fraction(1, 3)) + + PauliSum.term(PauliString({0: "Z", 1: "Z"}), Fraction(2, 5)) + ) + bound = computational_row_sum_bound(operator, 2) + norm = np.linalg.norm(to_dense(operator, 2), ord=2) + assert float(bound) >= norm + fast_bound = computational_row_taxicab_bound(operator, 2) + assert float(fast_bound) >= norm + collatz = collatz_taxicab_certificate(operator, 2, iterations=20) + assert float(collatz.bound) >= norm + assert collatz.bound <= fast_bound + + +def test_uniform_cluster_weights_reconstruct_orbit_density() -> None: + lattice = SquareLattice(4) + seed = PauliString({lattice.site(0, 0): "X", lattice.site(1, 0): "X"}) + _, members = canonical_translation(seed, lattice) + orbits = translation_orbits(PauliSum({member: 1 for member in members}), lattice) + certificate = build_uniform_cluster_certificate(orbits, 3, 2) + assert certificate.placements[0].weights == (Fraction(1, 4),) * 4 + assert certificate.row_sum_bound > 0 + + +def test_colored_orbit_placements_respect_unit_cell_stride() -> None: + lattice = SquareLattice(4) + seed = PauliString({lattice.site(1, 0): "X"}) + _, members = canonical_translation(seed, lattice, (2, 2)) + orbits = translation_orbits( + PauliSum({member: 1 for member in members}), + lattice, + unit_cell=(2, 2), + ) + certificate = build_uniform_cluster_certificate(orbits, 4, 3) + assert certificate.placements[0].offsets == ((0, 0), (2, 0), (0, 2), (2, 2)) + + +def test_partitioned_certificate_upper_bounds_small_operator_norm() -> None: + lattice = SquareLattice(4) + seed = PauliString({lattice.site(0, 0): "Z"}) + _, members = canonical_translation(seed, lattice, (2, 2)) + operator = PauliSum({member: 1 for member in members}) + certificate = build_partitioned_operator_certificate(operator, lattice) + assert certificate.global_bound == 4 + full = build_full_patch_operator_certificate(operator, lattice) + assert full.global_bound == 4 + phased = phase_partitioned_collatz_certificate( + operator, lattice, iterations=10 + ) + assert phased.global_bound == 4 diff --git a/tracks/qcs/solutions/WangTheoPhys/issue128/tests/test_cubic_field.py b/tracks/qcs/solutions/WangTheoPhys/issue128/tests/test_cubic_field.py new file mode 100644 index 000000000..ef42465d0 --- /dev/null +++ b/tracks/qcs/solutions/WangTheoPhys/issue128/tests/test_cubic_field.py @@ -0,0 +1,74 @@ +from __future__ import annotations + +from fractions import Fraction + +import pytest + +from trottercert.cubic_field import ( + Cubic, + fourth_order_suzuki_cubic_stages, +) +from trottercert.intervals import cube_root_four_interval +from trottercert.rigorous_fourth import fourth_order_suzuki_interval_stages + + +def test_cubic_multiplication_reduces_alpha_cubed() -> None: + alpha = Cubic(0, 1, 0) + assert alpha * alpha * alpha == Cubic(4, 0, 0) + assert alpha**4 == Cubic(0, 4, 0) + + +def test_exact_suzuki_scale_is_inverse_of_four_minus_alpha() -> None: + alpha = Cubic(0, 1, 0) + u = Cubic(Fraction(4, 15), Fraction(1, 15), Fraction(1, 60)) + assert u * (Cubic(4, 0, 0) - alpha) == Cubic.one() + + +def test_cubic_ring_arithmetic_and_rational_division() -> None: + left = Cubic(Fraction(1, 3), Fraction(2, 5), Fraction(-7, 11)) + right = Cubic(Fraction(-2, 7), Fraction(3, 13), Fraction(5, 17)) + assert left + right - right == left + assert 2 * left == left * 2 + assert left / 3 == Cubic( + Fraction(1, 9), Fraction(2, 15), Fraction(-7, 33) + ) + assert left**0 == Cubic.one() + with pytest.raises(ValueError, match="negative"): + _ = left**-1 + with pytest.raises(ZeroDivisionError): + _ = left / 0 + + +def test_exact_stages_enclose_inside_existing_interval_stages() -> None: + exact = fourth_order_suzuki_cubic_stages(4) + interval, root = fourth_order_suzuki_interval_stages( + 4, decimal_digits=24 + ) + assert len(exact) == len(interval) == 31 + for left, right in zip(exact, interval): + assert left.fragment_index == right.fragment_index + enclosed = left.coefficient.enclose(root) + assert right.coefficient.lower <= enclosed.lower + assert enclosed.upper <= right.coefficient.upper + + +def test_exact_stage_sum_is_one_for_every_fragment() -> None: + stages = fourth_order_suzuki_cubic_stages(4) + for fragment in range(4): + assert sum( + ( + stage.coefficient + for stage in stages + if stage.fragment_index == fragment + ), + Cubic.zero(), + ) == Cubic.one() + + +def test_cubic_interval_enclosure_contains_high_precision_root_value() -> None: + value = Cubic(Fraction(3, 7), Fraction(-5, 11), Fraction(13, 17)) + root = cube_root_four_interval(30) + interval = value.enclose(root) + midpoint = (root.lower + root.upper) / 2 + evaluated = value.a0 + value.a1 * midpoint + value.a2 * midpoint**2 + assert interval.lower <= evaluated <= interval.upper diff --git a/tracks/qcs/solutions/WangTheoPhys/issue128/tests/test_cubic_local.py b/tracks/qcs/solutions/WangTheoPhys/issue128/tests/test_cubic_local.py new file mode 100644 index 000000000..6b740759b --- /dev/null +++ b/tracks/qcs/solutions/WangTheoPhys/issue128/tests/test_cubic_local.py @@ -0,0 +1,30 @@ +from __future__ import annotations + +import pytest + +from trottercert.cubic_field import fourth_order_suzuki_cubic_stages +from trottercert.cubic_local import exact_log_e5_density +from trottercert.refined_error import certified_d4_cell_coefficients +from trottercert.rigorous_fourth import fourth_order_suzuki_interval_stages + + +@pytest.mark.slow +def test_exact_e5_is_enclosed_by_existing_interval_coefficients() -> None: + stages = fourth_order_suzuki_cubic_stages(4) + _, exact = exact_log_e5_density(stages) + interval_stages, root = fourth_order_suzuki_interval_stages(4) + expected = certified_d4_cell_coefficients(interval_stages) + observed = { + pauli: value.enclose(root) * 5 + for pauli, value in exact.items() + } + # The interval implementation can retain ghost terms whose exact cubic + # coefficients cancel to zero after dependency information is restored. + assert set(observed) <= set(expected) + assert all( + expected[pauli].lower + <= observed[pauli].lower + <= observed[pauli].upper + <= expected[pauli].upper + for pauli in observed + ) diff --git a/tracks/qcs/solutions/WangTheoPhys/issue128/tests/test_cubic_right_generator.py b/tracks/qcs/solutions/WangTheoPhys/issue128/tests/test_cubic_right_generator.py new file mode 100644 index 000000000..ffad8aa95 --- /dev/null +++ b/tracks/qcs/solutions/WangTheoPhys/issue128/tests/test_cubic_right_generator.py @@ -0,0 +1,31 @@ +from __future__ import annotations + +import pytest + +from trottercert.cubic_field import fourth_order_suzuki_cubic_stages +from trottercert.cubic_local import exact_right_generator_stage_contribution +from trottercert.hpc_artifacts import coordinate_encode_series, merge_coordinate_series + + +@pytest.mark.slow +def test_stage_contributions_reconstruct_fourth_order_conditions() -> None: + stages = fourth_order_suzuki_cubic_stages() + shards = [ + coordinate_encode_series( + *exact_right_generator_stage_contribution(stages, index, 4) + ) + for index in range(len(stages)) + ] + series = merge_coordinate_series(shards) + assert not series[1] + assert not series[2] + assert not series[3] + assert series[4] + + +def test_stage_contribution_rejects_invalid_index_and_order() -> None: + stages = fourth_order_suzuki_cubic_stages() + with pytest.raises(IndexError, match="stage index"): + exact_right_generator_stage_contribution(stages, len(stages), 4) + with pytest.raises(ValueError, match="nonnegative"): + exact_right_generator_stage_contribution(stages, 0, -1) diff --git a/tracks/qcs/solutions/WangTheoPhys/issue128/tests/test_defect_series.py b/tracks/qcs/solutions/WangTheoPhys/issue128/tests/test_defect_series.py new file mode 100644 index 000000000..8cd062f0b --- /dev/null +++ b/tracks/qcs/solutions/WangTheoPhys/issue128/tests/test_defect_series.py @@ -0,0 +1,9 @@ +from trottercert.defect_series import right_generator_local_series +from trottercert.higher_order import fourth_order_suzuki_stages + + +def test_suzuki_right_generator_vanishes_through_degree_three() -> None: + series = right_generator_local_series(fourth_order_suzuki_stages(4), 4) + for degree in (1, 2, 3): + assert sum(abs(value) for value in series[degree].values()) < 1e-9 + assert sum(abs(value) for value in series[4].values()) > 0 diff --git a/tracks/qcs/solutions/WangTheoPhys/issue128/tests/test_delivery_package.py b/tracks/qcs/solutions/WangTheoPhys/issue128/tests/test_delivery_package.py new file mode 100644 index 000000000..2a746710d --- /dev/null +++ b/tracks/qcs/solutions/WangTheoPhys/issue128/tests/test_delivery_package.py @@ -0,0 +1,77 @@ +from __future__ import annotations + +import json +from pathlib import Path +import sys + +import pytest + + +ROOT = Path(__file__).resolve().parents[1] +sys.path.insert(0, str(ROOT)) + +from scripts.package_delivery import ( # noqa: E402 + EXPECTED, + build_summary, + canonical_json, + render_summary_text, + require_equal, + write_delivery_files, +) + + +def test_build_summary_projects_frozen_certificate() -> None: + summary = build_summary(ROOT) + assert summary["schema_version"] == 1 + assert summary["benchmark"]["sites"] == 144 + assert summary["published_baseline"]["steps"] == 393 + assert summary["published_baseline"]["group_exponentials"] == 11_791 + assert summary["certified_result"]["steps"] == 97 + assert summary["certified_result"]["group_exponentials"] == 2_911 + assert summary["improvement"]["exact_ratio"] == [11_791, 2_911] + assert summary["improvement"]["decimal"] == "4.050498110614909" + assert summary["verification"]["d4_term_count"] == 75_324 + assert summary["verification"]["d4_group_count"] == 7_576 + assert summary["fivefold_followup"]["status"] == "not_certified" + assert summary["fivefold_followup"]["d5_term_count"] == 605_832 + assert summary["fivefold_followup"]["d5_group_count"] == 123_106 + + +def test_summary_keeps_outward_error_boundary() -> None: + result = build_summary(ROOT)["certified_result"] + assert result["error_upper_decimal_outward"] == "9.958938494314325e-7" + assert result["previous_step_error_upper_decimal_outward"] == "1.050565873970784e-6" + assert result["accepted_at_97"] is True + assert result["rejected_at_96"] is True + + +def test_summary_binds_frozen_digests() -> None: + sources = build_summary(ROOT)["sources"] + assert sources["main_certificate"]["sha256"] == EXPECTED["certificate_sha256"] + assert sources["d4_sidecar"]["sha256"] == EXPECTED["d4_sha256"] + assert sources["d5_sidecar"]["sha256"] == EXPECTED["d5_sha256"] + + +def test_require_equal_rejects_drift() -> None: + with pytest.raises(ValueError, match="candidate steps drift"): + require_equal("candidate steps", 96, 97) + + +def test_text_marks_fivefold_unproved() -> None: + text = render_summary_text(build_summary(ROOT)) + assert "CERTIFIED RESULT: 4.050498110614909x" in text + assert "FIVEFOLD STATUS: NOT CERTIFIED" in text + assert "No 78-step global error certificate is claimed or supplied." in text + assert "certified 5x" not in text.lower() + + +def test_canonical_json_is_sorted_and_newline_terminated() -> None: + assert canonical_json({"z": 1, "a": 2}) == '{\n "a": 2,\n "z": 1\n}\n' + + +def test_write_delivery_files_round_trips_summary(tmp_path: Path) -> None: + summary = build_summary(ROOT) + written = write_delivery_files(ROOT, tmp_path, capture=False) + assert written == {tmp_path / "issue128-summary.json", tmp_path / "issue128-summary.txt"} + assert json.loads((tmp_path / "issue128-summary.json").read_text()) == summary + assert (tmp_path / "issue128-summary.txt").read_text().endswith("\n") diff --git a/tracks/qcs/solutions/WangTheoPhys/issue128/tests/test_higher_order.py b/tracks/qcs/solutions/WangTheoPhys/issue128/tests/test_higher_order.py new file mode 100644 index 000000000..02fdf5ce3 --- /dev/null +++ b/tracks/qcs/solutions/WangTheoPhys/issue128/tests/test_higher_order.py @@ -0,0 +1,81 @@ +import mpmath as mp +import pytest + +from trottercert.algebra import PauliString, PauliSum +from trottercert.higher_order import ( + fourth_order_suzuki_stages, + higher_order_triangle_weights, + weak_compositions, + fourth_order_local_pauli_l1_density, + fourth_order_triple_jump_stages, + higher_order_combined_local_l1_density, + higher_order_combined_local_l1_density_symplectic, + _pairwise_anticommuting_float_bound, +) + + +def test_weak_compositions() -> None: + values = tuple(weak_compositions(3, 2)) + assert values == ((0, 3), (1, 2), (2, 1), (3, 0)) + + +def test_fourth_order_stage_consistency_and_palindrome() -> None: + stages = fourth_order_suzuki_stages(4) + assert len(stages) == 31 + assert [stage.fragment_index for stage in stages] == [ + stage.fragment_index for stage in reversed(stages) + ] + coefficients = [mp.mpf("0") for _ in range(4)] + for stage in stages: + coefficients[stage.fragment_index] += stage.coefficient + assert all(mp.almosteq(value, 1) for value in coefficients) + triple = fourth_order_triple_jump_stages(4) + assert len(triple) == 19 + assert [stage.fragment_index for stage in triple] == [ + stage.fragment_index for stage in reversed(triple) + ] + + +def test_higher_order_weights_are_positive_and_depth_five() -> None: + weights = higher_order_triangle_weights(fourth_order_suzuki_stages(2), 4) + assert weights + assert all(len(key) == 5 for key in weights) + assert all(value > 0 for value in weights.values()) + + +@pytest.mark.slow +def test_local_fourth_order_density_is_finite_and_positive() -> None: + density, _ = fourth_order_local_pauli_l1_density() + assert density > 0 + assert mp.isfinite(density) + + +@pytest.mark.slow +def test_combined_bj_bound_is_no_worse_than_expanded_bound() -> None: + stages = fourth_order_suzuki_stages(4) + combined = higher_order_combined_local_l1_density(stages, center=20) + assert combined > 0 + + +@pytest.mark.slow +def test_symplectic_combined_bound_matches_fraction_discovery() -> None: + stages = fourth_order_suzuki_stages(4) + fraction_value = higher_order_combined_local_l1_density(stages, center=2) + symplectic_value = higher_order_combined_local_l1_density_symplectic( + stages, + center=2, + ) + assert float(fraction_value) == pytest.approx(symplectic_value, rel=1e-12) + + +def test_pairwise_anticommuting_float_bound() -> None: + # X and Z anticommute, while I is left as a singleton. + paulis = ((1, 0), (0, 1), (0, 0)) + coefficients = mp.matrix([3.0, 4.0, 2.0]) + import numpy as np + + value = _pairwise_anticommuting_float_bound( + paulis, + np.asarray(coefficients, dtype=float).reshape(-1), + ) + assert value == pytest.approx(7.0) diff --git a/tracks/qcs/solutions/WangTheoPhys/issue128/tests/test_hpc_artifacts.py b/tracks/qcs/solutions/WangTheoPhys/issue128/tests/test_hpc_artifacts.py new file mode 100644 index 000000000..2abc07738 --- /dev/null +++ b/tracks/qcs/solutions/WangTheoPhys/issue128/tests/test_hpc_artifacts.py @@ -0,0 +1,64 @@ +from __future__ import annotations + +from fractions import Fraction +from pathlib import Path + +import pytest + +from trottercert.cubic_field import Cubic +from trottercert.hpc_artifacts import ( + coordinate_encode_terms, + read_shard_gzip, + sha256_file, + write_manifest_atomic, + write_shard_gzip, +) +from trottercert.local_commutators import CoordinateRegistry + + +def _fixture(order: tuple[tuple[int, int], ...]): + registry = CoordinateRegistry() + for coordinate in order: + registry.site(coordinate) + left = registry.site((0, 0)) + right = registry.site((1, 0)) + pauli = ((1 << left) | (1 << right), 1 << right) + return registry, {pauli: Cubic(Fraction(1, 3), Fraction(-2, 5), 7)} + + +def test_coordinate_encoding_is_registry_independent() -> None: + first = _fixture(((0, 0), (1, 0))) + second = _fixture(((1, 0), (0, 0))) + assert coordinate_encode_terms(*first) == coordinate_encode_terms(*second) + + +def test_canonical_gzip_and_manifest_are_deterministic(tmp_path: Path) -> None: + registry, terms = _fixture(((0, 0), (1, 0))) + payload = { + "schema_version": 1, + "kind": "test", + "terms": coordinate_encode_terms(registry, terms), + } + first = tmp_path / "first.json.gz" + second = tmp_path / "second.json.gz" + write_shard_gzip(first, payload) + write_shard_gzip(second, payload) + assert first.read_bytes() == second.read_bytes() + assert read_shard_gzip(first) == payload + assert sha256_file(first) == sha256_file(second) + + manifest = tmp_path / "manifest.json" + write_manifest_atomic(manifest, {"status": "complete", "count": 1}) + first_manifest = manifest.read_bytes() + write_manifest_atomic(manifest, {"status": "complete", "count": 1}) + assert manifest.read_bytes() == first_manifest + + +def test_corrupted_gzip_is_rejected(tmp_path: Path) -> None: + path = tmp_path / "shard.json.gz" + write_shard_gzip(path, {"schema_version": 1, "terms": []}) + raw = bytearray(path.read_bytes()) + raw[-8] ^= 1 + path.write_bytes(raw) + with pytest.raises(ValueError, match="gzip JSON"): + read_shard_gzip(path) diff --git a/tracks/qcs/solutions/WangTheoPhys/issue128/tests/test_intervals.py b/tracks/qcs/solutions/WangTheoPhys/issue128/tests/test_intervals.py new file mode 100644 index 000000000..c8dd2fb66 --- /dev/null +++ b/tracks/qcs/solutions/WangTheoPhys/issue128/tests/test_intervals.py @@ -0,0 +1,25 @@ +from fractions import Fraction + +from trottercert.intervals import ( + RationalInterval, + cube_root_four_interval, + outward_quantize, +) + + +def test_rational_interval_arithmetic_and_root_enclosure() -> None: + left = RationalInterval(Fraction(1, 3), Fraction(1, 2)) + right = RationalInterval(Fraction(-2), Fraction(-1)) + assert left + right == RationalInterval(Fraction(-5, 3), Fraction(-1, 2)) + assert left * right == RationalInterval(Fraction(-1), Fraction(-1, 3)) + root = cube_root_four_interval(12) + assert root.lower**3 <= 4 <= root.upper**3 + assert root.upper - root.lower == Fraction(1, 10**12) + + +def test_outward_quantize_contains_original_interval() -> None: + interval = RationalInterval(Fraction(1, 3), Fraction(5, 7)) + rounded = outward_quantize(interval, 10) + assert rounded.lower <= interval.lower + assert rounded.upper >= interval.upper + assert rounded == RationalInterval(Fraction(3, 10), Fraction(4, 5)) diff --git a/tracks/qcs/solutions/WangTheoPhys/issue128/tests/test_lattice_hamiltonian.py b/tracks/qcs/solutions/WangTheoPhys/issue128/tests/test_lattice_hamiltonian.py new file mode 100644 index 000000000..6e0bcb8bb --- /dev/null +++ b/tracks/qcs/solutions/WangTheoPhys/issue128/tests/test_lattice_hamiltonian.py @@ -0,0 +1,44 @@ +from fractions import Fraction + +import numpy as np + +from trottercert.algebra import commutator, to_dense +from trottercert.hamiltonian import ( + four_matching_fragments, + full_heisenberg_hamiltonian, + heisenberg_bond, +) +from trottercert.lattice import SquareLattice + + +def test_periodic_square_bond_and_matching_counts() -> None: + for length in (4, 6): + lattice = SquareLattice(length) + matchings = lattice.four_matchings() + assert len(lattice.bonds()) == 2 * length * length + assert all(len(group) == length * length // 2 for group in matchings) + assert set().union(*(set(group) for group in matchings)) == set(lattice.bonds()) + for group in matchings: + flattened = [site for bond in group for site in bond] + assert len(flattened) == len(set(flattened)) + + +def test_matching_fragments_reconstruct_hamiltonian() -> None: + lattice = SquareLattice(4) + total = sum(four_matching_fragments(lattice)[1:], four_matching_fragments(lattice)[0]) + assert total == full_heisenberg_hamiltonian(lattice) + + +def test_heisenberg_bond_spectrum() -> None: + values = np.linalg.eigvalsh(to_dense(heisenberg_bond(0, 1), 2)) + assert np.allclose(values, [-0.75, 0.25, 0.25, 0.25]) + + +def test_overlapping_bond_double_commutator_identity() -> None: + h12 = heisenberg_bond(0, 1) + h23 = heisenberg_bond(1, 2) + h13 = heisenberg_bond(0, 2) + lhs = commutator(h12, commutator(h12, h23)) + rhs = (h23 - h13).scale(Fraction(1, 2)) + assert lhs == rhs + assert np.allclose(to_dense(lhs, 3), to_dense(rhs, 3)) diff --git a/tracks/qcs/solutions/WangTheoPhys/issue128/tests/test_lie_series.py b/tracks/qcs/solutions/WangTheoPhys/issue128/tests/test_lie_series.py new file mode 100644 index 000000000..e8320d25a --- /dev/null +++ b/tracks/qcs/solutions/WangTheoPhys/issue128/tests/test_lie_series.py @@ -0,0 +1,25 @@ +import mpmath as mp + +from trottercert.higher_order import ( + fourth_order_suzuki_stages, + _second_order_scalar_stages, +) +from trottercert.lie_series import abstract_formula_log_series + + +def _degree_l1(series, degree: int) -> mp.mpf: + return sum(abs(value) for value in series[degree].values()) + + +def test_abstract_strang_has_zero_degree_two() -> None: + stages = _second_order_scalar_stages(4, mp.mpf(1)) + logarithm = abstract_formula_log_series(stages, 3) + assert _degree_l1(logarithm, 2) < mp.mpf("1e-60") + assert _degree_l1(logarithm, 3) > 0 + + +def test_abstract_suzuki_is_fourth_order() -> None: + logarithm = abstract_formula_log_series(fourth_order_suzuki_stages(4), 5) + for degree in (2, 3, 4): + assert _degree_l1(logarithm, degree) < mp.mpf("1e-60") + assert _degree_l1(logarithm, 5) > 0 diff --git a/tracks/qcs/solutions/WangTheoPhys/issue128/tests/test_local_commutators.py b/tracks/qcs/solutions/WangTheoPhys/issue128/tests/test_local_commutators.py new file mode 100644 index 000000000..0222f5f4c --- /dev/null +++ b/tracks/qcs/solutions/WangTheoPhys/issue128/tests/test_local_commutators.py @@ -0,0 +1,80 @@ +from fractions import Fraction + +from trottercert.baseline import strang_commutator_operators +from trottercert.hamiltonian import four_matching_fragments +from trottercert.lattice import SquareLattice +from trottercert.local_commutators import ( + DyadicLocalDensityEvaluator, + local_nested_pauli_l1_density, + matching_partner, + LocalDensityEvaluator, + SymplecticDyadicLocalDensityEvaluator, +) + + +def test_matching_partner_is_an_involution() -> None: + for color in range(4): + for coordinate in ((0, 0), (1, 0), (-1, 2), (3, -2)): + partner = matching_partner(coordinate, color) + assert matching_partner(partner, color) == coordinate + + +def test_local_depth_three_density_matches_unaliased_global_l6() -> None: + lattice = SquareLattice(6) + fragments = four_matching_fragments(lattice) + # Compare representative pure nested commutators, not the B_j sums. + for key in ((0, 0, 1), (2, 3, 0), (1, 2, 3)): + from trottercert.algebra import commutator + + global_operator = commutator( + fragments[key[0]], + commutator(fragments[key[1]], fragments[key[2]]), + ) + expected_density = global_operator.exact_real_l1() / 9 + assert local_nested_pauli_l1_density(key) == expected_density + + +def test_suffix_memoized_evaluator_matches_direct() -> None: + evaluator = LocalDensityEvaluator() + for key in ((0, 2, 3), (1, 2, 3), (0, 1, 2, 3)): + assert evaluator.pauli_l1_density(key) == local_nested_pauli_l1_density(key) + + +def test_dyadic_evaluator_matches_fraction_evaluator() -> None: + fraction_evaluator = LocalDensityEvaluator() + dyadic_evaluator = DyadicLocalDensityEvaluator() + for key in ( + (0,), + (1, 0), + (2, 1, 0), + (3, 2, 1, 0), + (0, 3, 2, 1, 0), + ): + fraction_operator = fraction_evaluator.evaluate(key) + dyadic_operator = dyadic_evaluator.evaluate(key) + exponent = dyadic_evaluator.denominator_exponent(key) + assert set(fraction_operator.terms) == set(dyadic_operator) + for pauli, coefficient in fraction_operator.terms.items(): + numerator = dyadic_operator[pauli] + assert coefficient.real == Fraction(numerator[0], 1 << exponent) + assert coefficient.imag == Fraction(numerator[1], 1 << exponent) + assert ( + dyadic_evaluator.pauli_l1_density(key) + == fraction_evaluator.pauli_l1_density(key) + ) + + +def test_symplectic_evaluator_matches_fraction_evaluator() -> None: + fraction_evaluator = LocalDensityEvaluator() + symplectic_evaluator = SymplecticDyadicLocalDensityEvaluator() + for key in ( + (0,), + (1, 0), + (2, 1, 0), + (3, 2, 1, 0), + (0, 3, 2, 1, 0), + ): + assert ( + symplectic_evaluator.pauli_l1_density(key) + == fraction_evaluator.pauli_l1_density(key) + ) diff --git a/tracks/qcs/solutions/WangTheoPhys/issue128/tests/test_matching_order_screen.py b/tracks/qcs/solutions/WangTheoPhys/issue128/tests/test_matching_order_screen.py new file mode 100644 index 000000000..0c16f2868 --- /dev/null +++ b/tracks/qcs/solutions/WangTheoPhys/issue128/tests/test_matching_order_screen.py @@ -0,0 +1,17 @@ +from __future__ import annotations + +import pytest + +from trottercert.matching_screen import permutation_from_index + + +def test_permutation_index_covers_exactly_24_orders() -> None: + values = [permutation_from_index(index) for index in range(24)] + assert len(set(values)) == 24 + assert values[0] == (0, 1, 2, 3) + assert values[-1] == (3, 2, 1, 0) + + +def test_permutation_index_rejects_out_of_range() -> None: + with pytest.raises(IndexError, match="permutation index"): + permutation_from_index(24) diff --git a/tracks/qcs/solutions/WangTheoPhys/issue128/tests/test_orbits.py b/tracks/qcs/solutions/WangTheoPhys/issue128/tests/test_orbits.py new file mode 100644 index 000000000..ed6a4f95f --- /dev/null +++ b/tracks/qcs/solutions/WangTheoPhys/issue128/tests/test_orbits.py @@ -0,0 +1,70 @@ +from fractions import Fraction + +from trottercert.algebra import PauliString, PauliSum, QComplex +from trottercert.lattice import SquareLattice +from trottercert.orbits import canonical_translation, translation_orbits + + +def test_wrapped_bond_gets_contiguous_representative() -> None: + lattice = SquareLattice(4) + pauli = PauliString({lattice.site(0, 0): "X", lattice.site(3, 0): "X"}) + representative, members = canonical_translation(pauli, lattice) + assert max(x for x, _, _ in representative) == 1 + assert len(members) == lattice.n_sites + + +def test_translation_invariant_orbit_decomposition() -> None: + lattice = SquareLattice(4) + seed = PauliString({lattice.site(0, 0): "X", lattice.site(1, 0): "X"}) + _, members = canonical_translation(seed, lattice) + operator = PauliSum({member: 2 for member in members}) + orbits = translation_orbits(operator, lattice) + assert len(orbits) == 1 + assert len(orbits[0].members) == lattice.n_sites + + +def test_noninvariant_operator_is_rejected() -> None: + lattice = SquareLattice(4) + operator = PauliSum.term(PauliString({0: "X"})) + try: + translation_orbits(operator, lattice) + except ValueError as error: + assert "not translation invariant" in str(error) + else: + raise AssertionError("expected a translation-invariance error") + + +def test_colored_two_by_two_unit_cell_orbit() -> None: + lattice = SquareLattice(4) + seed = PauliString({lattice.site(0, 0): "X"}) + _, members = canonical_translation(seed, lattice, (2, 2)) + operator = PauliSum({member: 1 for member in members}) + orbits = translation_orbits(operator, lattice, unit_cell=(2, 2)) + assert len(orbits) == 1 + assert len(orbits[0].members) == lattice.n_sites // 4 + + +def test_unit_cell_phase_is_preserved() -> None: + lattice = SquareLattice(4) + even = PauliString({lattice.site(0, 0): "X"}) + odd = PauliString({lattice.site(1, 0): "X"}) + even_rep, _ = canonical_translation(even, lattice, (2, 2)) + odd_rep, _ = canonical_translation(odd, lattice, (2, 2)) + assert even_rep != odd_rep + assert even_rep == ((0, 0, "X"),) + assert odd_rep == ((1, 0, "X"),) + + +def test_finite_size_stabilizer_scales_density_coefficient() -> None: + lattice = SquareLattice(4) + pauli = PauliString( + { + lattice.site(0, 0): "X", + lattice.site(2, 0): "X", + } + ) + _, members = canonical_translation(pauli, lattice, (2, 2)) + operator = PauliSum({member: 3 for member in members}) + orbit = translation_orbits(operator, lattice, unit_cell=(2, 2))[0] + assert orbit.stabilizer == 2 + assert orbit.coefficient == QComplex(Fraction(3, 2)) diff --git a/tracks/qcs/solutions/WangTheoPhys/issue128/tests/test_pareto.py b/tracks/qcs/solutions/WangTheoPhys/issue128/tests/test_pareto.py new file mode 100644 index 000000000..01a60684c --- /dev/null +++ b/tracks/qcs/solutions/WangTheoPhys/issue128/tests/test_pareto.py @@ -0,0 +1,23 @@ +from fractions import Fraction + +from trottercert.pareto import ( + minimum_published_suzuki_point, + recursive_suzuki_interval_stages, +) + + +def test_recursive_suzuki_stage_counts() -> None: + assert len(recursive_suzuki_interval_stages(2, decimal_digits=8)) == 7 + assert len(recursive_suzuki_interval_stages(4, decimal_digits=8)) == 31 + assert len(recursive_suzuki_interval_stages(6, decimal_digits=8)) == 151 + assert len(recursive_suzuki_interval_stages(8, decimal_digits=8)) == 751 + + +def test_small_published_pareto_search_is_minimal() -> None: + point = minimum_published_suzuki_point( + 4, + n_sites=4, + tolerance=Fraction(1, 100), + decimal_digits=8, + ) + assert point.global_error_bound <= Fraction(1, 100) diff --git a/tracks/qcs/solutions/WangTheoPhys/issue128/tests/test_processors.py b/tracks/qcs/solutions/WangTheoPhys/issue128/tests/test_processors.py new file mode 100644 index 000000000..e578b0eeb --- /dev/null +++ b/tracks/qcs/solutions/WangTheoPhys/issue128/tests/test_processors.py @@ -0,0 +1,49 @@ +import numpy as np + +from trottercert.algebra import QComplex, commutator, to_dense +from trottercert.formulas import leading_effective_error, strang_stages +from trottercert.hamiltonian import ( + four_matching_fragments, + full_heisenberg_hamiltonian, + heisenberg_bond, +) +from trottercert.lattice import SquareLattice +from trottercert.processors import ( + global_chirality_color_basis, + processor_images, + solve_processor_obstruction, + spin_chirality, +) + + +def test_spin_chirality_convention_and_hermiticity() -> None: + chi = spin_chirality(0, 1, 2) + lhs = commutator(heisenberg_bond(0, 1), heisenberg_bond(1, 2)) + assert lhs == chi.scale(QComplex(0, -1)) + assert chi.is_hermitian() + assert np.allclose(to_dense(chi, 3), to_dense(chi, 3).conj().T) + + +def test_global_color_chirality_basis_and_images_are_hermitian() -> None: + lattice = SquareLattice(4) + labeled = global_chirality_color_basis(lattice) + assert len(labeled) == 6 + assert all(operator.is_hermitian() for _, operator in labeled) + images = processor_images( + full_heisenberg_hamiltonian(lattice), + [operator for _, operator in labeled], + ) + assert all(image.is_hermitian() for image in images) + + +def test_projection_reconstructs_target_plus_residual() -> None: + lattice = SquareLattice(4) + labeled = global_chirality_color_basis(lattice) + images = processor_images( + full_heisenberg_hamiltonian(lattice), + [operator for _, operator in labeled], + ) + target = images[0].scale(2) - images[2] + result = solve_processor_obstruction(target, images) + assert result.exact_solution + assert not result.residual diff --git a/tracks/qcs/solutions/WangTheoPhys/issue128/tests/test_refined_error.py b/tracks/qcs/solutions/WangTheoPhys/issue128/tests/test_refined_error.py new file mode 100644 index 000000000..50fcd5ceb --- /dev/null +++ b/tracks/qcs/solutions/WangTheoPhys/issue128/tests/test_refined_error.py @@ -0,0 +1,190 @@ +from fractions import Fraction +from itertools import product + +import pytest + +from trottercert.algebra import ( + PauliString, + PauliSum, + commutator, + pauli_strings_commute, +) +from trottercert.hamiltonian import heisenberg_bond +from trottercert.local_commutators import CoordinateRegistry +from trottercert.refined_error import ( + HEISENBERG_BOND_PAULI_L1_GROWTH, + canonicalize_symplectic_unit_cell, + certified_d4_cell_coefficients, + certified_leading_e5_cell_l1, + defect_tail_site_bound, + symplectic_pauli_from_coordinates, +) +from trottercert.rigorous_fourth import fourth_order_suzuki_interval_stages + + +def test_defect_tail_decreases_with_steps() -> None: + stages, _ = fourth_order_suzuki_interval_stages(4, decimal_digits=8) + assert defect_tail_site_bound(stages, 200) < defect_tail_site_bound(stages, 100) + + +def test_grouped_d4_override_must_tighten_the_existing_constant() -> None: + from trottercert.refined_error import ( + build_refined_fourth_order_constants, + evaluate_refined_fourth_order_bound, + ) + + constants = build_refined_fourth_order_constants( + decimal_digits=8, + quantization_digits=8, + ) + tightened = constants.d4_site / 2 + original = evaluate_refined_fourth_order_bound(constants, 144, 116) + grouped = evaluate_refined_fourth_order_bound( + constants, + 144, + 116, + d4_site_override=tightened, + ) + assert grouped.degree_four_contribution == ( + original.degree_four_contribution / 2 + ) + with pytest.raises(ValueError, match="nonnegative"): + evaluate_refined_fourth_order_bound( + constants, + 144, + 116, + d4_site_override=Fraction(-1), + ) + with pytest.raises(ValueError, match="cannot exceed"): + evaluate_refined_fourth_order_bound( + constants, + 144, + 116, + d4_site_override=constants.d4_site + 1, + ) + + +def test_colored_unit_cell_canonicalization_merges_translates() -> None: + registry = CoordinateRegistry() + left = symplectic_pauli_from_coordinates( + registry, + ((0, 0, "X"), (1, 0, "X")), + ) + right = symplectic_pauli_from_coordinates( + registry, + ((2, 0, "X"), (3, 0, "X")), + ) + assert canonicalize_symplectic_unit_cell(registry, left) == ( + canonicalize_symplectic_unit_cell(registry, right) + ) + + +def test_heisenberg_bond_pauli_l1_growth_constant_is_exact() -> None: + bond_axes = tuple( + PauliString({0: axis, 1: axis}) + for axis in ("X", "Y", "Z") + ) + observed = [] + anticommuting_counts = [] + for left, right in product(("I", "X", "Y", "Z"), repeat=2): + pauli = PauliString({0: left, 1: right}) + anticommuting_counts.append( + sum( + not pauli_strings_commute(axis, pauli) + for axis in bond_axes + ) + ) + operator = PauliSum.term(pauli, Fraction(1)) + observed.append( + commutator( + heisenberg_bond(0, 1), + operator, + ).exact_axis_l1() + ) + assert set(anticommuting_counts) <= {0, 2} + assert max(observed) == HEISENBERG_BOND_PAULI_L1_GROWTH == 1 + + +@pytest.mark.slow +def test_canonical_d4_coefficients_are_valid_and_tighter() -> None: + stages, _ = fourth_order_suzuki_interval_stages( + 4, + decimal_digits=12, + ) + coefficients = certified_d4_cell_coefficients( + stages, + quantization_digits=18, + ) + assert coefficients + assert all( + interval.lower <= interval.upper + for interval in coefficients.values() + ) + assert sum( + (interval.abs_upper() for interval in coefficients.values()), + Fraction(), + ) <= 5 * certified_leading_e5_cell_l1( + stages, + quantization_digits=18, + ) + + +@pytest.mark.slow +def test_refined_bound_crosses_global_twofold_target() -> None: + from trottercert.refined_error import ( + build_refined_fourth_order_constants, + evaluate_refined_fourth_order_bound, + ) + + constants = build_refined_fourth_order_constants() + bound = evaluate_refined_fourth_order_bound(constants, 144, 116) + previous = evaluate_refined_fourth_order_bound(constants, 144, 115) + assert bound.global_error_bound <= Fraction(1, 10**6) + assert previous.global_error_bound > Fraction(1, 10**6) + + +@pytest.mark.slow +def test_grouped_d4_bound_crosses_global_fourfold_target() -> None: + from trottercert.anticommuting import ( + certify_anticommuting_partition, + discover_anticommuting_partition, + ) + from trottercert.refined_error import ( + build_refined_fourth_order_constants, + evaluate_refined_fourth_order_bound, + ) + + stages, _ = fourth_order_suzuki_interval_stages( + 4, + decimal_digits=12, + ) + coefficients = certified_d4_cell_coefficients( + stages, + quantization_digits=18, + ) + groups = discover_anticommuting_partition( + coefficients, + max_group_size=10, + ) + certificate = certify_anticommuting_partition( + coefficients, + groups, + ) + constants = build_refined_fourth_order_constants() + d4_site = certificate.bound / 4 + accepted = evaluate_refined_fourth_order_bound( + constants, + 144, + 97, + d4_site_override=d4_site, + ) + rejected = evaluate_refined_fourth_order_bound( + constants, + 144, + 96, + d4_site_override=d4_site, + ) + assert len(certificate.paulis) == 75_324 + assert len(certificate.groups) == 7_576 + assert accepted.global_error_bound <= Fraction(1, 10**6) + assert rejected.global_error_bound > Fraction(1, 10**6) diff --git a/tracks/qcs/solutions/WangTheoPhys/issue128/tests/test_representation.py b/tracks/qcs/solutions/WangTheoPhys/issue128/tests/test_representation.py new file mode 100644 index 000000000..9004f0b4d --- /dev/null +++ b/tracks/qcs/solutions/WangTheoPhys/issue128/tests/test_representation.py @@ -0,0 +1,11 @@ +import pytest + +from trottercert.representation import degree_three_four_matching_rank + + +@pytest.mark.slow +def test_degree_three_representation_has_full_free_lie_rank() -> None: + rank, columns, rows = degree_three_four_matching_rank(4) + assert rank == 20 + assert columns == 64 + assert rows == 4128 diff --git a/tracks/qcs/solutions/WangTheoPhys/issue128/tests/test_resources_fourth.py b/tracks/qcs/solutions/WangTheoPhys/issue128/tests/test_resources_fourth.py new file mode 100644 index 000000000..a2ce8ea81 --- /dev/null +++ b/tracks/qcs/solutions/WangTheoPhys/issue128/tests/test_resources_fourth.py @@ -0,0 +1,8 @@ +from trottercert.resources import fourth_order_four_matching_resources + + +def test_fourth_order_boundary_merging() -> None: + estimate = fourth_order_four_matching_resources(144, 59) + assert estimate.group_exponentials == 1771 + assert estimate.local_propagators == 127512 + assert estimate.cnot_upper == 382536 diff --git a/tracks/qcs/solutions/WangTheoPhys/issue128/tests/test_resources_verify.py b/tracks/qcs/solutions/WangTheoPhys/issue128/tests/test_resources_verify.py new file mode 100644 index 000000000..b81967352 --- /dev/null +++ b/tracks/qcs/solutions/WangTheoPhys/issue128/tests/test_resources_verify.py @@ -0,0 +1,159 @@ +import hashlib +import json +from fractions import Fraction +from pathlib import Path + +import pytest + +from trottercert.anticommuting import sqrt_fraction_upper +from trottercert.resources import ( + ceil_nth_root_fraction, + four_matching_resources, + required_steps, + symmetric_group_exponentials, + three_l_path_resources, +) +from trottercert.verify import _verify_d4_sidecar, verify_certificate + + +def test_integer_roots_and_stage_merging() -> None: + assert ceil_nth_root_fraction(Fraction(10), 2) == 4 + assert required_steps(Fraction(378), Fraction(1, 1000), 2) == 615 + assert symmetric_group_exponentials(4, 3) == 19 + + +def test_resource_counts() -> None: + baseline = four_matching_resources(144, 615) + candidate = three_l_path_resources(144, 559) + assert baseline.local_propagators == 265_752 + assert candidate.local_propagators == 107_376 + + +def test_verifier_recomputes_and_rejects_corruption(tmp_path) -> None: + certificate = { + "schema_version": 1, + "benchmark": { + "normalization": "(XX+YY+ZZ)/4", + "length": 12, + "tolerance": [1, 1000], + }, + "proof": { + "reference_length": 6, + "baseline_density": [21, 8], + "candidate_density": [13, 6], + }, + "claimed_resources": { + "baseline_steps": 615, + "candidate_steps": 559, + "baseline_local_propagators": 265752, + "candidate_local_propagators": 107376, + "baseline_cnot_upper": 797256, + "candidate_cnot_upper": 966384, + }, + } + path = tmp_path / "certificate.json" + path.write_text(json.dumps(certificate)) + result = verify_certificate(path) + assert result["twofold_local_target_met"] + assert not result["twofold_compiled_cnot_target_met"] + certificate["proof"]["candidate_density"] = [2, 1] + path.write_text(json.dumps(certificate)) + try: + verify_certificate(path) + except ValueError as error: + assert "candidate density" in str(error) + else: + raise AssertionError("corrupted certificate was accepted") + + +def _write_synthetic_d4_sidecar( + root: Path, + payload: dict[str, object], + *, + digest: str | None = None, +) -> tuple[Path, dict[str, object]]: + sidecar = root / "d4.json" + raw = ( + json.dumps(payload, sort_keys=True, separators=(",", ":")) + + "\n" + ).encode() + sidecar.write_bytes(raw) + main = root / "main.json" + main.write_text("{}") + metadata = { + "path": sidecar.name, + "sha256": ( + hashlib.sha256(raw).hexdigest() + if digest is None + else digest + ), + "max_group_size": 3, + "cell_norm_upper": payload["cell_bound"], + } + return main, {"d4_certificate": metadata} + + +def _synthetic_d4_payload() -> dict[str, object]: + bound = sqrt_fraction_upper(Fraction(3)) + return { + "schema_version": 1, + "coefficient_denominator": 1, + "terms": [ + [0, 1, 1, 1], + [1, 0, 1, 1], + [1, 1, 1, 1], + ], + "sqrt_denominator": bound.denominator, + "groups": [ + [[0, 1, 2], bound.numerator], + ], + "cell_bound": [bound.numerator, bound.denominator], + } + + +def test_d4_sidecar_rejects_coverage_corruption(tmp_path) -> None: + payload = _synthetic_d4_payload() + payload["groups"][0][0].pop() + main, candidate = _write_synthetic_d4_sidecar(tmp_path, payload) + with pytest.raises(ValueError, match="coverage"): + _verify_d4_sidecar(main, candidate) + + +def test_d4_sidecar_rejects_commuting_group(tmp_path) -> None: + payload = _synthetic_d4_payload() + payload["terms"][2][:2] = [2, 0] + main, candidate = _write_synthetic_d4_sidecar(tmp_path, payload) + with pytest.raises(ValueError, match="anticommute"): + _verify_d4_sidecar(main, candidate) + + +def test_d4_sidecar_rejects_reduced_group_bound(tmp_path) -> None: + payload = _synthetic_d4_payload() + payload["groups"][0][1] -= 1 + main, candidate = _write_synthetic_d4_sidecar(tmp_path, payload) + with pytest.raises(ValueError, match="group bound"): + _verify_d4_sidecar(main, candidate) + + +def test_d4_sidecar_rejects_digest_corruption(tmp_path) -> None: + payload = _synthetic_d4_payload() + main, candidate = _write_synthetic_d4_sidecar( + tmp_path, + payload, + digest="0" * 64, + ) + with pytest.raises(ValueError, match="sidecar digest"): + _verify_d4_sidecar(main, candidate) + + +def test_v3_deliverable_fast_verifies() -> None: + root = Path(__file__).resolve().parents[1] + result = verify_certificate( + root / "certificates" / "issue128-certificate.json" + ) + assert result["global_twofold_target_met"] + assert result["global_fourfold_target_met"] + assert result["candidate_steps"] == 97 + assert result["published_steps"] == 393 + assert result["d4_term_count"] == 75_324 + assert result["d4_group_count"] == 7_576 diff --git a/tracks/qcs/solutions/WangTheoPhys/issue128/tests/test_rigorous_fourth.py b/tracks/qcs/solutions/WangTheoPhys/issue128/tests/test_rigorous_fourth.py new file mode 100644 index 000000000..ca9e7da31 --- /dev/null +++ b/tracks/qcs/solutions/WangTheoPhys/issue128/tests/test_rigorous_fourth.py @@ -0,0 +1,35 @@ +from fractions import Fraction + +import pytest + +from trottercert.rigorous_fourth import _rational_pair_bound +from trottercert.intervals import RationalInterval + + +def test_rational_pair_bound_dominates_euclidean_norm() -> None: + coefficients = ( + RationalInterval.point(3), + RationalInterval.point(4), + RationalInterval(Fraction(19, 10), Fraction(21, 10)), + ) + bound = _rational_pair_bound(coefficients, ((0, 1),), (2,)) + assert float(bound) >= (3**2 + 4**2) ** 0.5 + 2.1 + + +@pytest.mark.slow +def test_fourth_order_rational_certificate_crosses_target() -> None: + from trottercert.rigorous_fourth import fourth_order_rational_pair_certificate + + certificate = fourth_order_rational_pair_certificate(center=17) + assert certificate.site_density_upper < Fraction(9687, 100) + + +@pytest.mark.slow +def test_published_triangle_certificate_reproduces_baseline() -> None: + from trottercert.rigorous_fourth import ( + fourth_order_published_triangle_certificate, + ) + + certificate = fourth_order_published_triangle_certificate(decimal_digits=12) + assert certificate.center == 20 + assert Fraction(164) < certificate.site_density_upper < Fraction(166) diff --git a/tracks/qcs/solutions/WangTheoPhys/issue128/tests/test_series_formulas.py b/tracks/qcs/solutions/WangTheoPhys/issue128/tests/test_series_formulas.py new file mode 100644 index 000000000..a0788e45f --- /dev/null +++ b/tracks/qcs/solutions/WangTheoPhys/issue128/tests/test_series_formulas.py @@ -0,0 +1,56 @@ +from fractions import Fraction + +import numpy as np +from scipy.linalg import expm, logm + +from trottercert.algebra import PauliString, PauliSum, to_dense +from trottercert.formulas import ( + formula_log_series, + formula_log_through_degree_three, + leading_effective_error, + strang_stages, +) +from trottercert.series import exponential_series, logarithm_series + + +def pauli(site: int, op: str) -> PauliSum: + return PauliSum.term(PauliString({site: op})) + + +def test_exponential_and_logarithm_series_are_inverse() -> None: + x = pauli(0, "X") + series = exponential_series(x, Fraction(1, 3), 5) + recovered = logarithm_series(series) + assert recovered.coefficients[1] == x.scale(Fraction(1, 3)) + assert all(not recovered.coefficients[k] for k in range(2, 6)) + + +def test_strang_log_has_only_odd_terms_through_fourth_order() -> None: + fragments = (pauli(0, "X"), pauli(0, "Z")) + coefficients = formula_log_series(strang_stages(fragments), order=4) + assert not coefficients[2] + assert not coefficients[4] + + +def test_leading_error_matches_small_time_matrix_logarithm() -> None: + fragments = (pauli(0, "X"), pauli(0, "Z")) + stages = strang_stages(fragments) + e3 = leading_effective_error(stages) + h = to_dense(fragments[0] + fragments[1], 1) + predicted = to_dense(e3, 1) + for time in (1e-2, 5e-3): + product = np.eye(2, dtype=complex) + for stage in stages: + product = product @ expm( + -1j * float(stage.coefficient) * time * to_dense(stage.fragment, 1) + ) + effective = 1j * logm(product) / time + residual = (effective - h) / (time**2) + assert np.allclose(residual, predicted, atol=2e-4) + + +def test_graded_bch_matches_generic_series() -> None: + stages = strang_stages((pauli(0, "X"), pauli(0, "Z"))) + generic = formula_log_series(stages, order=3) + graded = formula_log_through_degree_three(stages) + assert tuple(generic[1:4]) == graded diff --git a/tracks/qcs/solutions/WangTheoPhys/issue128/tests/test_small_crosscheck.py b/tracks/qcs/solutions/WangTheoPhys/issue128/tests/test_small_crosscheck.py new file mode 100644 index 000000000..cfbfec7f9 --- /dev/null +++ b/tracks/qcs/solutions/WangTheoPhys/issue128/tests/test_small_crosscheck.py @@ -0,0 +1,10 @@ +from fractions import Fraction + +from trottercert.crosscheck import small_exact_crosscheck + + +def test_small_exact_evolution_is_below_certificate() -> None: + result = small_exact_crosscheck(2, Fraction(1, 1000)) + assert result["bound_dominates_empirical_error"] + assert result["bound_meets_tolerance"] + assert result["steps"] == 97 diff --git a/tracks/qcs/solutions/WangTheoPhys/issue128/tests/test_su2clusters.py b/tracks/qcs/solutions/WangTheoPhys/issue128/tests/test_su2clusters.py new file mode 100644 index 000000000..94b63cc77 --- /dev/null +++ b/tracks/qcs/solutions/WangTheoPhys/issue128/tests/test_su2clusters.py @@ -0,0 +1,35 @@ +from trottercert.hamiltonian import full_heisenberg_hamiltonian +from trottercert.lattice import SquareLattice +from trottercert.su2clusters import ( + three_color_l_path_clusters, + three_l_path_fragments, +) + + +def test_three_color_decomposition_requires_compatible_torus() -> None: + try: + three_color_l_path_clusters(SquareLattice(10)) + except ValueError as error: + assert "requires 3 | L" in str(error) + else: + raise AssertionError("expected incompatible periodic coloring to fail") + + +def test_same_color_clusters_are_site_disjoint_l6() -> None: + lattice = SquareLattice(6) + groups = three_color_l_path_clusters(lattice) + assert all(len(group) == lattice.n_sites // 3 for group in groups) + for group in groups: + all_sites = [site for cluster in group for site in cluster.sites] + assert len(all_sites) == len(set(all_sites)) + + +def test_three_color_clusters_cover_every_bond_once() -> None: + lattice = SquareLattice(6) + clusters = three_color_l_path_clusters(lattice) + covered = [bond for group in clusters for cluster in group for bond in cluster.bonds] + assert len(covered) == len(set(covered)) + assert set(covered) == set(lattice.bonds()) + assert sum(three_l_path_fragments(lattice)[1:], three_l_path_fragments(lattice)[0]) == ( + full_heisenberg_hamiltonian(lattice) + ) diff --git a/tracks/qcs/solutions/WangTheoPhys/issue128/tests/test_support_groups.py b/tracks/qcs/solutions/WangTheoPhys/issue128/tests/test_support_groups.py new file mode 100644 index 000000000..32ba0d86e --- /dev/null +++ b/tracks/qcs/solutions/WangTheoPhys/issue128/tests/test_support_groups.py @@ -0,0 +1,94 @@ +from __future__ import annotations + +from fractions import Fraction + +import pytest + +from trottercert.intervals import RationalInterval +from trottercert.support_groups import ( + build_d5_payload, + canonical_gzip_bytes, + certify_support_partition, + decode_d5_gzip, + discover_support_partition, + verify_d5_payload, +) + + +def test_support_certificate_rejects_cross_support_pair() -> None: + coefficients = { + (1, 0): RationalInterval.point(1), + (2, 0): RationalInterval.point(1), + } + with pytest.raises(ValueError, match="same support"): + certify_support_partition(coefficients, (((1, 0), (2, 0)),)) + + +def test_support_certificate_rejects_commuting_pair() -> None: + coefficients = { + (3, 0): RationalInterval.point(1), + (0, 3): RationalInterval.point(1), + } + with pytest.raises(ValueError, match="anticommute"): + certify_support_partition(coefficients, (((3, 0), (0, 3)),)) + + +def test_discovery_and_certification_cover_each_term_once() -> None: + coefficients = { + (3, 0): RationalInterval.point(Fraction(3, 7)), + (1, 2): RationalInterval.point(Fraction(-2, 9)), + (2, 1): RationalInterval.point(Fraction(5, 11)), + (4, 0): RationalInterval.point(Fraction(1, 13)), + } + groups = discover_support_partition(coefficients, max_group_size=3) + certificate = certify_support_partition(coefficients, groups) + flattened = [pauli for group in groups for pauli in group] + assert sorted(flattened) == sorted(coefficients) + assert certificate.bound <= sum( + (coefficient.abs_upper() for coefficient in coefficients.values()), + Fraction(), + ) + + +def test_discovery_scores_require_exact_coverage() -> None: + coefficients = { + (1, 2): RationalInterval.point(1), + (2, 1): RationalInterval.point(1), + } + with pytest.raises(ValueError, match="scores.*cover"): + discover_support_partition( + coefficients, + discovery_scores={(1, 2): Fraction(1)}, + ) + + +def test_support_certificate_rejects_missing_coverage() -> None: + coefficients = { + (1, 2): RationalInterval.point(1), + (2, 1): RationalInterval.point(1), + } + with pytest.raises(ValueError, match="coverage"): + certify_support_partition(coefficients, (((1, 2),),)) + + +def test_d5_sidecar_is_deterministic_and_rejects_missing_term() -> None: + coefficients = { + (1, 2): RationalInterval.point(Fraction(3, 7)), + (2, 1): RationalInterval.point(Fraction(-2, 9)), + } + first = build_d5_payload(coefficients) + second = build_d5_payload(coefficients) + encoded = canonical_gzip_bytes(first) + assert encoded == canonical_gzip_bytes(second) + assert encoded[9] == 0x13 + assert decode_d5_gzip(encoded) == first + verify_d5_payload(first) + groups = first["groups"] + assert isinstance(groups, list) + first_group = groups[0] + assert isinstance(first_group, list) + indices = first_group[0] + assert isinstance(indices, list) + indices.pop() + with pytest.raises(ValueError, match="coverage"): + verify_d5_payload(first)