From f3046a3e03f0b301870c6446101122f384cd4c22 Mon Sep 17 00:00:00 2001 From: JunkaiWang-TheoPhy <1181100960@qq.com> Date: Wed, 29 Jul 2026 06:17:03 +0800 Subject: [PATCH 01/27] certify tighter Trotter bounds for issue 128 --- tracks/qcs/solutions/WangTheoPhys/README.md | 182 ++++++ .../certificates/issue128-certificate.json | 132 +++++ .../certificates/issue128-degree3-rank.json | 10 + .../issue128-small-crosscheck.json | 11 + .../issue128-three-cluster-certificate.json | 42 ++ .../certificates/issue128-verification.txt | 83 +++ .../WangTheoPhys/issue128/pyproject.toml | 22 + .../issue128/scripts/build_v3_certificate.py | 180 ++++++ .../issue128/scripts/crosscheck_small.py | 25 + .../WangTheoPhys/issue128/scripts/verify.py | 28 + .../issue128/src/trottercert/__init__.py | 5 + .../issue128/src/trottercert/algebra.py | 308 +++++++++++ .../issue128/src/trottercert/baseline.py | 48 ++ .../issue128/src/trottercert/clusters.py | 520 ++++++++++++++++++ .../issue128/src/trottercert/crosscheck.py | 62 +++ .../issue128/src/trottercert/defect_series.py | 153 ++++++ .../issue128/src/trottercert/formulas.py | 94 ++++ .../issue128/src/trottercert/hamiltonian.py | 28 + .../issue128/src/trottercert/higher_order.py | 516 +++++++++++++++++ .../issue128/src/trottercert/intervals.py | 146 +++++ .../issue128/src/trottercert/lattice.py | 55 ++ .../issue128/src/trottercert/lie_series.py | 124 +++++ .../src/trottercert/local_commutators.py | 397 +++++++++++++ .../issue128/src/trottercert/orbits.py | 137 +++++ .../issue128/src/trottercert/pareto.py | 150 +++++ .../issue128/src/trottercert/processors.py | 166 ++++++ .../issue128/src/trottercert/refined_error.py | 296 ++++++++++ .../src/trottercert/representation.py | 36 ++ .../issue128/src/trottercert/resources.py | 83 +++ .../src/trottercert/rigorous_fourth.py | 424 ++++++++++++++ .../issue128/src/trottercert/series.py | 107 ++++ .../issue128/src/trottercert/su2clusters.py | 70 +++ .../issue128/src/trottercert/verify.py | 393 +++++++++++++ .../issue128/tests/test_algebra.py | 48 ++ .../issue128/tests/test_baseline.py | 30 + .../issue128/tests/test_clusters.py | 90 +++ .../issue128/tests/test_defect_series.py | 9 + .../issue128/tests/test_higher_order.py | 81 +++ .../issue128/tests/test_intervals.py | 25 + .../tests/test_lattice_hamiltonian.py | 44 ++ .../issue128/tests/test_lie_series.py | 25 + .../issue128/tests/test_local_commutators.py | 80 +++ .../issue128/tests/test_orbits.py | 70 +++ .../issue128/tests/test_pareto.py | 23 + .../issue128/tests/test_processors.py | 49 ++ .../issue128/tests/test_refined_error.py | 25 + .../issue128/tests/test_representation.py | 11 + .../issue128/tests/test_resources_fourth.py | 8 + .../issue128/tests/test_resources_verify.py | 72 +++ .../issue128/tests/test_rigorous_fourth.py | 35 ++ .../issue128/tests/test_series_formulas.py | 56 ++ .../issue128/tests/test_small_crosscheck.py | 9 + .../issue128/tests/test_su2clusters.py | 35 ++ 53 files changed, 5858 insertions(+) create mode 100644 tracks/qcs/solutions/WangTheoPhys/README.md create mode 100644 tracks/qcs/solutions/WangTheoPhys/issue128/certificates/issue128-certificate.json create mode 100644 tracks/qcs/solutions/WangTheoPhys/issue128/certificates/issue128-degree3-rank.json create mode 100644 tracks/qcs/solutions/WangTheoPhys/issue128/certificates/issue128-small-crosscheck.json create mode 100644 tracks/qcs/solutions/WangTheoPhys/issue128/certificates/issue128-three-cluster-certificate.json create mode 100644 tracks/qcs/solutions/WangTheoPhys/issue128/certificates/issue128-verification.txt create mode 100644 tracks/qcs/solutions/WangTheoPhys/issue128/pyproject.toml create mode 100644 tracks/qcs/solutions/WangTheoPhys/issue128/scripts/build_v3_certificate.py create mode 100644 tracks/qcs/solutions/WangTheoPhys/issue128/scripts/crosscheck_small.py create mode 100644 tracks/qcs/solutions/WangTheoPhys/issue128/scripts/verify.py create mode 100644 tracks/qcs/solutions/WangTheoPhys/issue128/src/trottercert/__init__.py create mode 100644 tracks/qcs/solutions/WangTheoPhys/issue128/src/trottercert/algebra.py create mode 100644 tracks/qcs/solutions/WangTheoPhys/issue128/src/trottercert/baseline.py create mode 100644 tracks/qcs/solutions/WangTheoPhys/issue128/src/trottercert/clusters.py create mode 100644 tracks/qcs/solutions/WangTheoPhys/issue128/src/trottercert/crosscheck.py create mode 100644 tracks/qcs/solutions/WangTheoPhys/issue128/src/trottercert/defect_series.py create mode 100644 tracks/qcs/solutions/WangTheoPhys/issue128/src/trottercert/formulas.py create mode 100644 tracks/qcs/solutions/WangTheoPhys/issue128/src/trottercert/hamiltonian.py create mode 100644 tracks/qcs/solutions/WangTheoPhys/issue128/src/trottercert/higher_order.py create mode 100644 tracks/qcs/solutions/WangTheoPhys/issue128/src/trottercert/intervals.py create mode 100644 tracks/qcs/solutions/WangTheoPhys/issue128/src/trottercert/lattice.py create mode 100644 tracks/qcs/solutions/WangTheoPhys/issue128/src/trottercert/lie_series.py create mode 100644 tracks/qcs/solutions/WangTheoPhys/issue128/src/trottercert/local_commutators.py create mode 100644 tracks/qcs/solutions/WangTheoPhys/issue128/src/trottercert/orbits.py create mode 100644 tracks/qcs/solutions/WangTheoPhys/issue128/src/trottercert/pareto.py create mode 100644 tracks/qcs/solutions/WangTheoPhys/issue128/src/trottercert/processors.py create mode 100644 tracks/qcs/solutions/WangTheoPhys/issue128/src/trottercert/refined_error.py create mode 100644 tracks/qcs/solutions/WangTheoPhys/issue128/src/trottercert/representation.py create mode 100644 tracks/qcs/solutions/WangTheoPhys/issue128/src/trottercert/resources.py create mode 100644 tracks/qcs/solutions/WangTheoPhys/issue128/src/trottercert/rigorous_fourth.py create mode 100644 tracks/qcs/solutions/WangTheoPhys/issue128/src/trottercert/series.py create mode 100644 tracks/qcs/solutions/WangTheoPhys/issue128/src/trottercert/su2clusters.py create mode 100644 tracks/qcs/solutions/WangTheoPhys/issue128/src/trottercert/verify.py create mode 100644 tracks/qcs/solutions/WangTheoPhys/issue128/tests/test_algebra.py create mode 100644 tracks/qcs/solutions/WangTheoPhys/issue128/tests/test_baseline.py create mode 100644 tracks/qcs/solutions/WangTheoPhys/issue128/tests/test_clusters.py create mode 100644 tracks/qcs/solutions/WangTheoPhys/issue128/tests/test_defect_series.py create mode 100644 tracks/qcs/solutions/WangTheoPhys/issue128/tests/test_higher_order.py create mode 100644 tracks/qcs/solutions/WangTheoPhys/issue128/tests/test_intervals.py create mode 100644 tracks/qcs/solutions/WangTheoPhys/issue128/tests/test_lattice_hamiltonian.py create mode 100644 tracks/qcs/solutions/WangTheoPhys/issue128/tests/test_lie_series.py create mode 100644 tracks/qcs/solutions/WangTheoPhys/issue128/tests/test_local_commutators.py create mode 100644 tracks/qcs/solutions/WangTheoPhys/issue128/tests/test_orbits.py create mode 100644 tracks/qcs/solutions/WangTheoPhys/issue128/tests/test_pareto.py create mode 100644 tracks/qcs/solutions/WangTheoPhys/issue128/tests/test_processors.py create mode 100644 tracks/qcs/solutions/WangTheoPhys/issue128/tests/test_refined_error.py create mode 100644 tracks/qcs/solutions/WangTheoPhys/issue128/tests/test_representation.py create mode 100644 tracks/qcs/solutions/WangTheoPhys/issue128/tests/test_resources_fourth.py create mode 100644 tracks/qcs/solutions/WangTheoPhys/issue128/tests/test_resources_verify.py create mode 100644 tracks/qcs/solutions/WangTheoPhys/issue128/tests/test_rigorous_fourth.py create mode 100644 tracks/qcs/solutions/WangTheoPhys/issue128/tests/test_series_formulas.py create mode 100644 tracks/qcs/solutions/WangTheoPhys/issue128/tests/test_small_crosscheck.py create mode 100644 tracks/qcs/solutions/WangTheoPhys/issue128/tests/test_su2clusters.py diff --git a/tracks/qcs/solutions/WangTheoPhys/README.md b/tracks/qcs/solutions/WangTheoPhys/README.md new file mode 100644 index 000000000..4d440060c --- /dev/null +++ b/tracks/qcs/solutions/WangTheoPhys/README.md @@ -0,0 +1,182 @@ +# 🔭 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 136 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 | 136 | +| Merged group exponentials | 11,791 | 4,081 | +| Bond propagators | 848,952 | 293,832 | +| CNOT upper bound (3 per bond) | 2,546,856 | 881,496 | + +The exact resource improvement is + +```text +11791 / 4081 = 2.889242832639059... +``` + +At 136 steps the certified global operator-norm error is +`9.910917811853245e-7`. At 135 steps the bound is +`1.033576466170403e-6`, so 136 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. converts the logarithm into the exact right generator; and +6. closes every degree at finite step size with a rational geometric locality + tail. + +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 136 steps + +| Right-generator degree | Global contribution | +|---:|---:| +| 4 | `4.340985168056117e-7` | +| 5 | `3.830281030637750e-8` | +| 6 | `2.230486607644384e-7` | +| 7 | `2.933476798355164e-8` | +| 8 and above | `2.663070253253453e-7` | +| **Total** | **`9.910917811853245e-7`** | + +All certificate values are stored as exact rational numerators and +denominators in +[`issue128/certificates/issue128-certificate.json`](issue128/certificates/issue128-certificate.json). + +## 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); +- E. Schubert and C. B. Mendl, + [*Trotter error bounds 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 +with a generic locality-tail theorem. Their merged group counts are 269,677, +36,361, 83,101, and 377,251 respectively, so simply raising the recursive +Suzuki order does not provide a closer rigorous competitor at this benchmark. + +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 deep verifier regenerates the published baseline, local degree-five + coefficient, degree-seven majorant, finite tail, integer step boundary, and + resource arithmetic. +- A degenerate 2x2 periodic-algebra sanity check gives actual spectral-norm + error `2.3625152178806053e-10`, below its outward-rounded certificate + `2.7555555555555555e-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. + +## Reproduce + +From this directory: + +```bash +python -m pip install -e '.[test]' +pytest -q +PYTHONPATH=src python scripts/verify.py \ + certificates/issue128-certificate.json +``` + +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 +``` + +The default test suite excludes explicitly marked research-reproduction tests. +The submitted run completed with 57 passing tests; the deep verifier also +completed successfully. + +## Files + +```text +issue128/ +├── certificates/ exact certificate and verification summaries +├── 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 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2, + "steps": 44946, + "stages_per_step": 7, + "group_exponentials": 269677, + "global_error_upper_decimal_summary": "9.9999203748666826e-07" + }, + { + "order": 4, + "steps": 1212, + "stages_per_step": 31, + "group_exponentials": 36361, + "global_error_upper_decimal_summary": "9.9682621337975879e-07" + }, + { + "order": 6, + "steps": 554, + "stages_per_step": 151, + "group_exponentials": 83101, + "global_error_upper_decimal_summary": "9.9542219002167203e-07" + }, + { + "order": 8, + "steps": 503, + "stages_per_step": 751, + "group_exponentials": 377251, + "global_error_upper_decimal_summary": "9.902910692421506e-07" + } + ], + "claims": { + "global_twofold_target_met": true, + "exact_improvement_ratio": [ + 11791, + 4081 + ] + } +} 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..6a1d72a38 --- /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=136", + "steps": 136, + "empirical_operator_norm_error": 2.3625152178806053e-10, + "certified_upper_bound": 2.7555555555555555e-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..3abb3915b --- /dev/null +++ b/tracks/qcs/solutions/WangTheoPhys/issue128/certificates/issue128-verification.txt @@ -0,0 +1,83 @@ +Issue #128 deep verification transcript +======================================= + +Status +------ +valid: true +verification_level: deep +deep_proof_regenerated: true +test_suite: 56 passed, 7 slow tests deselected + +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 + E7 majorant + exact right-generator tail +certified E5 site Pauli-l1 upper bound: 1.0312915120959002 +certified E7 site majorant: 9652.49828556245 + +At r=136: +degree-4 contribution: 4.340985168056117e-7 +degree-5 contribution: 3.830281030637750e-8 +degree-6 contribution: 2.230486607644384e-7 +degree-7 contribution: 2.933476798355164e-8 +degree >= 8 tail: 2.663070253253453e-7 +global error upper bound: 9.910917811853245e-7 + +At r=135: +global error upper bound: 1.033576466170403e-6 + +Therefore r=136 is the smallest step count accepted by this certificate. + +Candidate resources +------------------- +steps: 136 +group exponentials: 4081 +bond propagators: 293832 +CNOT upper bound: 881496 + +Certified comparison +-------------------- +exact group/CNOT improvement ratio: 11791/4081 +decimal improvement: 2.889242832639059 +global twofold 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/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/build_v3_certificate.py b/tracks/qcs/solutions/WangTheoPhys/issue128/scripts/build_v3_certificate.py new file mode 100644 index 000000000..0840838b5 --- /dev/null +++ b/tracks/qcs/solutions/WangTheoPhys/issue128/scripts/build_v3_certificate.py @@ -0,0 +1,180 @@ +#!/usr/bin/env python3 +from __future__ import annotations + +import json +from fractions import Fraction +from pathlib import Path + +from trottercert.pareto import minimum_published_suzuki_point +from trottercert.refined_error import ( + build_refined_fourth_order_constants, + 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" + + +def pair(value: Fraction) -> list[int]: + return [value.numerator, value.denominator] + + +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, + ) + low = 1 + high = published_steps + while low < high: + middle = (low + high) // 2 + if ( + evaluate_refined_fourth_order_bound( + constants, + n_sites, + middle, + ).global_error_bound + <= tolerance + ): + high = middle + else: + low = middle + 1 + candidate_steps = low + candidate = evaluate_refined_fourth_order_bound( + constants, + n_sites, + candidate_steps, + ) + previous = evaluate_refined_fourth_order_bound( + constants, + n_sites, + candidate_steps - 1, + ) + 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_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), + "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, + "exact_improvement_ratio": pair( + Fraction(published_groups, candidate_groups) + ), + }, + } + OUTPUT.write_text(json.dumps(certificate, indent=2) + "\n") + 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/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/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..8383f23a9 --- /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=136. +# The exact rational value remains in certificates/issue128-certificate.json. +REFINED_BOUND_N144_R136 = Fraction(992, 10**9) +REFINED_STEPS = 136 + + +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_R136 * Fraction(n_sites, 144)) + return { + "length": length, + "n_sites": n_sites, + "model_note": ( + "degenerate 2x2 periodic algebra sanity check for schema-v3 r=136" + 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/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/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/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..b8538ceae --- /dev/null +++ b/tracks/qcs/solutions/WangTheoPhys/issue128/src/trottercert/refined_error.py @@ -0,0 +1,296 @@ +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 SymplecticDyadicLocalDensityEvaluator +from .rigorous_fourth import ( + IntervalStage, + fourth_order_suzuki_interval_stages, +) + + +IntervalWord = tuple[int, ...] +IntervalWordSeries = list[dict[IntervalWord, RationalInterval]] + + +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 certified_leading_e5_cell_l1( + stages: Sequence[IntervalStage], + *, + quantization_digits: int = 24, +) -> Fraction: + 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) + coefficients: dict[tuple[int, int], tuple[int, int]] = {} + for word, (word_lower, word_upper) in word_endpoints.items(): + for pauli, numerator in evaluator.evaluate(word).items(): + 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 + coefficients[pauli] = (lower, upper) + numerator = sum(max(abs(lower), abs(upper)) for lower, upper in coefficients.values()) + # Dynkin projection divides by degree five; local depth-five coefficients + # have common dyadic denominator 2**6. + return Fraction(numerator, grid * 5 * (1 << 6)) + + +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 (3/2)*s at support size s=2,...,7. + nested_max = Fraction(3, 2) + for support_size in range(2, 8): + nested_max *= Fraction(3 * support_size, 2) + 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) * (3/2)**q * (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 = Fraction(3, 2) * 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 lattice bonds; + # each bond commutator multiplies l1 by at most 3/2. + h_e5 = 36 * e5_site + hh_e5 = 42 * h_e5 + hhh_e5 = 48 * hh_e5 + h_e7 = 48 * 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, +) -> RefinedFourthOrderBound: + + def contribution(degree: int, density: Fraction) -> Fraction: + return Fraction(n_sites) * density / ( + (degree + 1) * steps**degree + ) + + c4 = contribution(4, constants.d4_site) + 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/verify.py b/tracks/qcs/solutions/WangTheoPhys/issue128/src/trottercert/verify.py new file mode 100644 index 000000000..45f5d42c3 --- /dev/null +++ b/tracks/qcs/solutions/WangTheoPhys/issue128/src/trottercert/verify.py @@ -0,0 +1,393 @@ +from __future__ import annotations + +import json +from fractions import Fraction +from pathlib import Path + +from .baseline import pauli_l1_second_order_constant +from .hamiltonian import four_matching_fragments +from .lattice import SquareLattice +from .resources import ( + fourth_order_four_matching_resources, + four_matching_resources, + required_steps, + three_l_path_resources, +) +from .intervals import cube_root_four_interval +from .su2clusters import three_l_path_fragments + + +EXPECTED_NORMALIZATION = "(XX+YY+ZZ)/4" + + +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_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, +) -> 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_plus_E7_majorant_plus_exact_generator_tail" + ): + raise ValueError("candidate structure 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") + + 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, + 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") + rebuilt = evaluate_refined_fourth_order_bound( + constants, + n_sites, + candidate_steps, + ) + rebuilt_previous = evaluate_refined_fourth_order_bound( + constants, + n_sites, + candidate_steps - 1, + ) + 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") + 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), + "exact_improvement_ratio": str(ratio), + "improvement": float(ratio), + "global_twofold_target_met": ratio >= 2, + } + + +def verify_certificate( + path: str | Path, + *, + deep: bool = False, +) -> dict[str, object]: + data = json.loads(Path(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) + 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_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_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_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_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_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..09f241ce6 --- /dev/null +++ b/tracks/qcs/solutions/WangTheoPhys/issue128/tests/test_refined_error.py @@ -0,0 +1,25 @@ +from fractions import Fraction + +import pytest + +from trottercert.refined_error import defect_tail_site_bound +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) + + +@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, 136) + previous = evaluate_refined_fourth_order_bound(constants, 144, 135) + assert bound.global_error_bound <= Fraction(1, 10**6) + assert previous.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..3a6a45ea3 --- /dev/null +++ b/tracks/qcs/solutions/WangTheoPhys/issue128/tests/test_resources_verify.py @@ -0,0 +1,72 @@ +import json +from fractions import Fraction +from pathlib import Path + +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_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 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["candidate_steps"] == 136 + assert result["published_steps"] == 393 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..879a0be4b --- /dev/null +++ b/tracks/qcs/solutions/WangTheoPhys/issue128/tests/test_small_crosscheck.py @@ -0,0 +1,9 @@ +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"] 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) + ) From f4e6913bc8ebb71e2fcdc4c400c9f6b901ddb6db Mon Sep 17 00:00:00 2001 From: JunkaiWang-TheoPhy <1181100960@qq.com> Date: Wed, 29 Jul 2026 18:00:53 +0800 Subject: [PATCH 02/27] tighten Heisenberg commutator growth bound --- tracks/qcs/solutions/WangTheoPhys/README.md | 50 +++++++++------ .../certificates/issue128-certificate.json | 64 +++++++++---------- .../issue128-small-crosscheck.json | 8 +-- .../certificates/issue128-verification.txt | 38 +++++------ .../issue128/src/trottercert/crosscheck.py | 10 +-- .../issue128/src/trottercert/refined_error.py | 31 ++++++--- .../issue128/tests/test_refined_error.py | 43 ++++++++++++- .../issue128/tests/test_resources_verify.py | 2 +- 8 files changed, 154 insertions(+), 92 deletions(-) diff --git a/tracks/qcs/solutions/WangTheoPhys/README.md b/tracks/qcs/solutions/WangTheoPhys/README.md index 4d440060c..884b80f3b 100644 --- a/tracks/qcs/solutions/WangTheoPhys/README.md +++ b/tracks/qcs/solutions/WangTheoPhys/README.md @@ -18,26 +18,26 @@ 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 136 Trotter +we certify the same five-copy fourth-order Suzuki circuit with 116 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 | 136 | -| Merged group exponentials | 11,791 | 4,081 | -| Bond propagators | 848,952 | 293,832 | -| CNOT upper bound (3 per bond) | 2,546,856 | 881,496 | +| Trotter steps | 393 | 116 | +| Merged group exponentials | 11,791 | 3,481 | +| Bond propagators | 848,952 | 250,632 | +| CNOT upper bound (3 per bond) | 2,546,856 | 751,896 | The exact resource improvement is ```text -11791 / 4081 = 2.889242832639059... +11791 / 3481 = 3.3872450445274347... ``` -At 136 steps the certified global operator-norm error is -`9.910917811853245e-7`. At 135 steps the bound is -`1.033576466170403e-6`, so 136 is the smallest integer accepted by this +At 116 steps the certified global operator-norm error is +`9.72530161861392e-7`. At 115 steps the bound is +`1.0097845503628042e-6`, so 116 is the smallest integer accepted by this certificate. ## What is new @@ -60,6 +60,18 @@ Our proof instead: 6. closes every degree at finite step size with a rational geometric locality tail. +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 @@ -81,16 +93,16 @@ 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 136 steps +## Error ledger at 116 steps | Right-generator degree | Global contribution | |---:|---:| -| 4 | `4.340985168056117e-7` | -| 5 | `3.830281030637750e-8` | -| 6 | `2.230486607644384e-7` | -| 7 | `2.933476798355164e-8` | -| 8 and above | `2.663070253253453e-7` | -| **Total** | **`9.910917811853245e-7`** | +| 4 | `8.201852948883738e-7` | +| 5 | `5.6564503095749916e-8` | +| 6 | `5.398589151384083e-8` | +| 7 | `5.4166008370568104e-9` | +| 8 and above | `3.637787152637068e-8` | +| **Total** | **`9.72530161861392e-7`** | All certificate values are stored as exact rational numerators and denominators in @@ -127,8 +139,8 @@ their empirical constants cannot replace the pinned rigorous baseline. coefficient, degree-seven majorant, finite tail, integer step boundary, and resource arithmetic. - A degenerate 2x2 periodic-algebra sanity check gives actual spectral-norm - error `2.3625152178806053e-10`, below its outward-rounded certificate - `2.7555555555555555e-8`. + error `4.463500595491358e-10`, below its outward-rounded certificate + `2.7027777777777777e-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 @@ -163,7 +175,7 @@ PYTHONPATH=src python scripts/build_v3_certificate.py ``` The default test suite excludes explicitly marked research-reproduction tests. -The submitted run completed with 57 passing tests; the deep verifier also +The submitted run completed with 58 passing tests; the deep verifier also completed successfully. ## Files diff --git a/tracks/qcs/solutions/WangTheoPhys/issue128/certificates/issue128-certificate.json b/tracks/qcs/solutions/WangTheoPhys/issue128/certificates/issue128-certificate.json index d73c72b01..ea176b867 100644 --- a/tracks/qcs/solutions/WangTheoPhys/issue128/certificates/issue128-certificate.json +++ b/tracks/qcs/solutions/WangTheoPhys/issue128/certificates/issue128-certificate.json @@ -41,40 +41,40 @@ 640000000000000000000 ], "e7_site_majorant": [ - 138378215421823266251416809140316312512990788121212758221898482890582557837721563811695446577, - 14336000000000000000000000000000000000000000000000000000000000000000000000000000000000000 + 189819225544339185530064210069021004818917404830195827464881320837561807733500087533189913, + 224000000000000000000000000000000000000000000000000000000000000000000000000000000000000 ], - 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"candidate_steps": 136, + "candidate_steps": 116, "published_group_exponentials": 11791, - "candidate_group_exponentials": 4081, + "candidate_group_exponentials": 3481, "published_bond_propagators": 848952, - "candidate_bond_propagators": 293832, + "candidate_bond_propagators": 250632, "published_cnot_upper": 2546856, - "candidate_cnot_upper": 881496 + "candidate_cnot_upper": 751896 }, "published_recursive_suzuki_audit": [ { "order": 2, - "steps": 44946, + "steps": 29964, "stages_per_step": 7, - "group_exponentials": 269677, + "group_exponentials": 179785, "global_error_upper_decimal_summary": "9.9999203748666826e-07" }, { "order": 4, - "steps": 1212, + "steps": 808, "stages_per_step": 31, - "group_exponentials": 36361, + "group_exponentials": 24241, "global_error_upper_decimal_summary": "9.9682621337975879e-07" }, { "order": 6, - "steps": 554, + "steps": 370, "stages_per_step": 151, - "group_exponentials": 83101, - "global_error_upper_decimal_summary": "9.9542219002167203e-07" + "group_exponentials": 55501, + "global_error_upper_decimal_summary": "9.8463753882538689e-07" }, { "order": 8, - "steps": 503, + "steps": 335, "stages_per_step": 751, - "group_exponentials": 377251, - "global_error_upper_decimal_summary": "9.902910692421506e-07" + "group_exponentials": 251251, + "global_error_upper_decimal_summary": "9.9829085642259163e-07" } ], "claims": { "global_twofold_target_met": true, "exact_improvement_ratio": [ 11791, - 4081 + 3481 ] } } diff --git a/tracks/qcs/solutions/WangTheoPhys/issue128/certificates/issue128-small-crosscheck.json b/tracks/qcs/solutions/WangTheoPhys/issue128/certificates/issue128-small-crosscheck.json index 6a1d72a38..9779f2e94 100644 --- a/tracks/qcs/solutions/WangTheoPhys/issue128/certificates/issue128-small-crosscheck.json +++ b/tracks/qcs/solutions/WangTheoPhys/issue128/certificates/issue128-small-crosscheck.json @@ -1,10 +1,10 @@ { "length": 2, "n_sites": 4, - "model_note": "degenerate 2x2 periodic algebra sanity check for schema-v3 r=136", - "steps": 136, - "empirical_operator_norm_error": 2.3625152178806053e-10, - "certified_upper_bound": 2.7555555555555555e-08, + "model_note": "degenerate 2x2 periodic algebra sanity check for schema-v3 r=116", + "steps": 116, + "empirical_operator_norm_error": 4.463500595491358e-10, + "certified_upper_bound": 2.7027777777777777e-08, "requested_tolerance": 1e-06, "bound_dominates_empirical_error": true, "bound_meets_tolerance": true diff --git a/tracks/qcs/solutions/WangTheoPhys/issue128/certificates/issue128-verification.txt b/tracks/qcs/solutions/WangTheoPhys/issue128/certificates/issue128-verification.txt index 3abb3915b..66c1e47af 100644 --- a/tracks/qcs/solutions/WangTheoPhys/issue128/certificates/issue128-verification.txt +++ b/tracks/qcs/solutions/WangTheoPhys/issue128/certificates/issue128-verification.txt @@ -6,7 +6,7 @@ Status valid: true verification_level: deep deep_proof_regenerated: true -test_suite: 56 passed, 7 slow tests deselected +test_suite: 58 passed, 7 slow tests deselected Benchmark --------- @@ -35,32 +35,34 @@ Candidate certificate formula: the same five-copy fourth-order Suzuki formula method: local log E5 + E7 majorant + exact right-generator tail certified E5 site Pauli-l1 upper bound: 1.0312915120959002 -certified E7 site majorant: 9652.49828556245 +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 -At r=136: -degree-4 contribution: 4.340985168056117e-7 -degree-5 contribution: 3.830281030637750e-8 -degree-6 contribution: 2.230486607644384e-7 -degree-7 contribution: 2.933476798355164e-8 -degree >= 8 tail: 2.663070253253453e-7 -global error upper bound: 9.910917811853245e-7 +At r=116: +degree-4 contribution: 8.201852948883738e-7 +degree-5 contribution: 5.6564503095749916e-8 +degree-6 contribution: 5.398589151384083e-8 +degree-7 contribution: 5.4166008370568104e-9 +degree >= 8 tail: 3.637787152637068e-8 +global error upper bound: 9.72530161861392e-7 -At r=135: -global error upper bound: 1.033576466170403e-6 +At r=115: +global error upper bound: 1.0097845503628042e-6 -Therefore r=136 is the smallest step count accepted by this certificate. +Therefore r=116 is the smallest step count accepted by this certificate. Candidate resources ------------------- -steps: 136 -group exponentials: 4081 -bond propagators: 293832 -CNOT upper bound: 881496 +steps: 116 +group exponentials: 3481 +bond propagators: 250632 +CNOT upper bound: 751896 Certified comparison -------------------- -exact group/CNOT improvement ratio: 11791/4081 -decimal improvement: 2.889242832639059 +exact group/CNOT improvement ratio: 11791/3481 +decimal improvement: 3.3872450445274347 global twofold target met: true Additional published recursive-Suzuki audit diff --git a/tracks/qcs/solutions/WangTheoPhys/issue128/src/trottercert/crosscheck.py b/tracks/qcs/solutions/WangTheoPhys/issue128/src/trottercert/crosscheck.py index 8383f23a9..523e76659 100644 --- a/tracks/qcs/solutions/WangTheoPhys/issue128/src/trottercert/crosscheck.py +++ b/tracks/qcs/solutions/WangTheoPhys/issue128/src/trottercert/crosscheck.py @@ -9,10 +9,10 @@ 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=136. +# A short outward-rounded summary of the schema-v3 bound at N=144, r=116. # The exact rational value remains in certificates/issue128-certificate.json. -REFINED_BOUND_N144_R136 = Fraction(992, 10**9) -REFINED_STEPS = 136 +REFINED_BOUND_N144 = Fraction(973, 10**9) +REFINED_STEPS = 116 def small_exact_crosscheck( @@ -44,12 +44,12 @@ def small_exact_crosscheck( 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_R136 * Fraction(n_sites, 144)) + 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=136" + "degenerate 2x2 periodic algebra sanity check for schema-v3 r=116" if length == 2 else "periodic square lattice" ), diff --git a/tracks/qcs/solutions/WangTheoPhys/issue128/src/trottercert/refined_error.py b/tracks/qcs/solutions/WangTheoPhys/issue128/src/trottercert/refined_error.py index b8538ceae..baa23223e 100644 --- a/tracks/qcs/solutions/WangTheoPhys/issue128/src/trottercert/refined_error.py +++ b/tracks/qcs/solutions/WangTheoPhys/issue128/src/trottercert/refined_error.py @@ -16,6 +16,11 @@ 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 _interval_series_identity(order: int) -> IntervalWordSeries: result = [{} for _ in range(order + 1)] @@ -153,10 +158,14 @@ def certified_e7_cell_l1_majorant( Fraction(), ) # A length-seven right-nested commutator starts from the 3/2 l1 cell - # density and gains at most (3/2)*s at support size s=2,...,7. + # 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 *= Fraction(3 * support_size, 2) + nested_max *= ( + HEISENBERG_BOND_PAULI_L1_GROWTH * support_size + ) return word_l1 * nested_max / 7 @@ -175,7 +184,7 @@ def defect_tail_site_bound( A cell starts with Pauli-l1 density 3/2. After ``q`` local adjoints, support counting gives at most - (3/2) * (3/2)**q * (q+1)! + (3/2) * (q+1)! per cell. The conjugation Taylor coefficient contributes ``1/q!``; the Duhamel time integral contributes ``1/(q+1)``. Those factorial @@ -187,7 +196,9 @@ def defect_tail_site_bound( total = Fraction() for stage in stages: coefficient = _abs_upper(stage.coefficient) - ratio = Fraction(3, 2) * prefix / steps + 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) @@ -235,12 +246,12 @@ def build_refined_fourth_order_constants( ) / 4 e7_site = certified_e7_cell_l1_majorant(stages) / 4 - # ad_H on an s-site Pauli term has at most 4s overlapping lattice bonds; - # each bond commutator multiplies l1 by at most 3/2. - h_e5 = 36 * e5_site - hh_e5 = 42 * h_e5 - hhh_e5 = 48 * hh_e5 - h_e7 = 48 * e7_site + # 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), diff --git a/tracks/qcs/solutions/WangTheoPhys/issue128/tests/test_refined_error.py b/tracks/qcs/solutions/WangTheoPhys/issue128/tests/test_refined_error.py index 09f241ce6..e03923371 100644 --- a/tracks/qcs/solutions/WangTheoPhys/issue128/tests/test_refined_error.py +++ b/tracks/qcs/solutions/WangTheoPhys/issue128/tests/test_refined_error.py @@ -1,8 +1,19 @@ from fractions import Fraction +from itertools import product import pytest -from trottercert.refined_error import defect_tail_site_bound +from trottercert.algebra import ( + PauliString, + PauliSum, + commutator, + pauli_strings_commute, +) +from trottercert.hamiltonian import heisenberg_bond +from trottercert.refined_error import ( + HEISENBERG_BOND_PAULI_L1_GROWTH, + defect_tail_site_bound, +) from trottercert.rigorous_fourth import fourth_order_suzuki_interval_stages @@ -11,6 +22,32 @@ def test_defect_tail_decreases_with_steps() -> None: assert defect_tail_site_bound(stages, 200) < defect_tail_site_bound(stages, 100) +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_refined_bound_crosses_global_twofold_target() -> None: from trottercert.refined_error import ( @@ -19,7 +56,7 @@ def test_refined_bound_crosses_global_twofold_target() -> None: ) constants = build_refined_fourth_order_constants() - bound = evaluate_refined_fourth_order_bound(constants, 144, 136) - previous = evaluate_refined_fourth_order_bound(constants, 144, 135) + 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) diff --git a/tracks/qcs/solutions/WangTheoPhys/issue128/tests/test_resources_verify.py b/tracks/qcs/solutions/WangTheoPhys/issue128/tests/test_resources_verify.py index 3a6a45ea3..b9c670573 100644 --- a/tracks/qcs/solutions/WangTheoPhys/issue128/tests/test_resources_verify.py +++ b/tracks/qcs/solutions/WangTheoPhys/issue128/tests/test_resources_verify.py @@ -68,5 +68,5 @@ def test_v3_deliverable_fast_verifies() -> None: root / "certificates" / "issue128-certificate.json" ) assert result["global_twofold_target_met"] - assert result["candidate_steps"] == 136 + assert result["candidate_steps"] == 116 assert result["published_steps"] == 393 From b42c1065b7dc4221a9c5b156c64aeea12e09b891 Mon Sep 17 00:00:00 2001 From: JunkaiWang-TheoPhy <1181100960@qq.com> Date: Wed, 29 Jul 2026 21:29:43 +0800 Subject: [PATCH 03/27] document 4x anticommuting certificate design --- ...-29-anticommuting-d4-certificate-design.md | 164 ++++++++++++++++++ 1 file changed, 164 insertions(+) create mode 100644 tracks/qcs/solutions/WangTheoPhys/issue128/docs/plans/2026-07-29-anticommuting-d4-certificate-design.md 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. From d2331e2965b5d110d70fb828b4f0d2ac901ebfc6 Mon Sep 17 00:00:00 2001 From: JunkaiWang-TheoPhy <1181100960@qq.com> Date: Wed, 29 Jul 2026 21:34:15 +0800 Subject: [PATCH 04/27] plan 4x anticommuting certificate implementation --- ...2026-07-29-anticommuting-d4-certificate.md | 948 ++++++++++++++++++ 1 file changed, 948 insertions(+) create mode 100644 tracks/qcs/solutions/WangTheoPhys/issue128/docs/superpowers/plans/2026-07-29-anticommuting-d4-certificate.md 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. From bab62e092bec0c4eb9262fb4c8fecd4e33876029 Mon Sep 17 00:00:00 2001 From: JunkaiWang-TheoPhy <1181100960@qq.com> Date: Wed, 29 Jul 2026 21:36:51 +0800 Subject: [PATCH 05/27] expose canonical interval D4 coefficients --- .../issue128/src/trottercert/refined_error.py | 120 ++++++++++++++++-- .../issue128/tests/test_refined_error.py | 44 +++++++ 2 files changed, 155 insertions(+), 9 deletions(-) diff --git a/tracks/qcs/solutions/WangTheoPhys/issue128/src/trottercert/refined_error.py b/tracks/qcs/solutions/WangTheoPhys/issue128/src/trottercert/refined_error.py index baa23223e..fdac0753d 100644 --- a/tracks/qcs/solutions/WangTheoPhys/issue128/src/trottercert/refined_error.py +++ b/tracks/qcs/solutions/WangTheoPhys/issue128/src/trottercert/refined_error.py @@ -6,7 +6,11 @@ from typing import Sequence from .intervals import RationalInterval, outward_quantize -from .local_commutators import SymplecticDyadicLocalDensityEvaluator +from .local_commutators import ( + CoordinateRegistry, + SymplecticDyadicLocalDensityEvaluator, + SymplecticPauli, +) from .rigorous_fourth import ( IntervalStage, fourth_order_suzuki_interval_stages, @@ -22,6 +26,57 @@ 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) @@ -116,11 +171,12 @@ def interval_formula_log_series( return logarithm -def certified_leading_e5_cell_l1( +def _leading_e5_cell_interval_coefficients( stages: Sequence[IntervalStage], *, - quantization_digits: int = 24, -) -> Fraction: + 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]] = {} @@ -132,9 +188,15 @@ def certified_leading_e5_cell_l1( ) evaluator = SymplecticDyadicLocalDensityEvaluator(shared_coordinates=True) - coefficients: dict[tuple[int, int], tuple[int, int]] = {} + registry = evaluator.registries[0] + coefficients: dict[SymplecticPauli, tuple[int, int]] = {} for word, (word_lower, word_upper) in word_endpoints.items(): - for pauli, numerator in evaluator.evaluate(word).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 @@ -142,11 +204,51 @@ def certified_leading_e5_cell_l1( else: lower += numerator * word_upper upper += numerator * word_lower - coefficients[pauli] = (lower, upper) - numerator = sum(max(abs(lower), abs(upper)) for lower, upper in coefficients.values()) + 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. - return Fraction(numerator, grid * 5 * (1 << 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( diff --git a/tracks/qcs/solutions/WangTheoPhys/issue128/tests/test_refined_error.py b/tracks/qcs/solutions/WangTheoPhys/issue128/tests/test_refined_error.py index e03923371..801b4b7ff 100644 --- a/tracks/qcs/solutions/WangTheoPhys/issue128/tests/test_refined_error.py +++ b/tracks/qcs/solutions/WangTheoPhys/issue128/tests/test_refined_error.py @@ -10,9 +10,14 @@ 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 @@ -22,6 +27,21 @@ def test_defect_tail_decreases_with_steps() -> None: assert defect_tail_site_bound(stages, 200) < defect_tail_site_bound(stages, 100) +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}) @@ -48,6 +68,30 @@ def test_heisenberg_bond_pauli_l1_growth_constant_is_exact() -> None: 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 ( From c0543db10a590c6eb8b5e41454c5dbb89ee4e207 Mon Sep 17 00:00:00 2001 From: JunkaiWang-TheoPhy <1181100960@qq.com> Date: Wed, 29 Jul 2026 21:39:33 +0800 Subject: [PATCH 06/27] certify anticommuting Pauli partitions --- .../issue128/src/trottercert/anticommuting.py | 149 ++++++++++++++++++ .../issue128/tests/test_anticommuting.py | 90 +++++++++++ 2 files changed, 239 insertions(+) create mode 100644 tracks/qcs/solutions/WangTheoPhys/issue128/src/trottercert/anticommuting.py create mode 100644 tracks/qcs/solutions/WangTheoPhys/issue128/tests/test_anticommuting.py 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/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)) From fdbaa1b2bbd11b4cb80c1cf42853b391e4201f89 Mon Sep 17 00:00:00 2001 From: JunkaiWang-TheoPhy <1181100960@qq.com> Date: Wed, 29 Jul 2026 21:46:41 +0800 Subject: [PATCH 07/27] apply grouped D4 error certificate --- .../issue128/src/trottercert/refined_error.py | 18 +++- .../issue128/tests/test_refined_error.py | 84 +++++++++++++++++++ 2 files changed, 101 insertions(+), 1 deletion(-) diff --git a/tracks/qcs/solutions/WangTheoPhys/issue128/src/trottercert/refined_error.py b/tracks/qcs/solutions/WangTheoPhys/issue128/src/trottercert/refined_error.py index fdac0753d..da2f7a5d7 100644 --- a/tracks/qcs/solutions/WangTheoPhys/issue128/src/trottercert/refined_error.py +++ b/tracks/qcs/solutions/WangTheoPhys/issue128/src/trottercert/refined_error.py @@ -370,14 +370,30 @@ 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) + 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) diff --git a/tracks/qcs/solutions/WangTheoPhys/issue128/tests/test_refined_error.py b/tracks/qcs/solutions/WangTheoPhys/issue128/tests/test_refined_error.py index 801b4b7ff..50fcd5ceb 100644 --- a/tracks/qcs/solutions/WangTheoPhys/issue128/tests/test_refined_error.py +++ b/tracks/qcs/solutions/WangTheoPhys/issue128/tests/test_refined_error.py @@ -27,6 +27,43 @@ def test_defect_tail_decreases_with_steps() -> None: 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( @@ -104,3 +141,50 @@ def test_refined_bound_crosses_global_twofold_target() -> None: 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) From d7381a630b1e42194fb6635ffb0864ca326db0e2 Mon Sep 17 00:00:00 2001 From: JunkaiWang-TheoPhy <1181100960@qq.com> Date: Wed, 29 Jul 2026 21:54:48 +0800 Subject: [PATCH 08/27] add machine-checkable grouped D4 certificate --- .../certificates/issue128-certificate.json | 50 +++--- .../certificates/issue128-d4-groups.json | 1 + .../issue128/scripts/build_v3_certificate.py | 105 ++++++++++++- .../issue128/src/trottercert/verify.py | 145 +++++++++++++++++- .../issue128/tests/test_resources_verify.py | 91 ++++++++++- 5 files changed, 366 insertions(+), 26 deletions(-) create mode 100644 tracks/qcs/solutions/WangTheoPhys/issue128/certificates/issue128-d4-groups.json diff --git a/tracks/qcs/solutions/WangTheoPhys/issue128/certificates/issue128-certificate.json b/tracks/qcs/solutions/WangTheoPhys/issue128/certificates/issue128-certificate.json index ea176b867..c988161f9 100644 --- a/tracks/qcs/solutions/WangTheoPhys/issue128/certificates/issue128-certificate.json +++ b/tracks/qcs/solutions/WangTheoPhys/issue128/certificates/issue128-certificate.json @@ -33,7 +33,7 @@ }, "candidate": { "formula": "five_copy_suzuki_fourth_order", - "proof_method": "local_log_E5_plus_E7_majorant_plus_exact_generator_tail", + "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": [ @@ -44,37 +44,48 @@ 189819225544339185530064210069021004818917404830195827464881320837561807733500087533189913, 224000000000000000000000000000000000000000000000000000000000000000000000000000000000000 ], - "steps": 116, - "group_exponentials": 3481, + "d4_certificate": { + "path": "issue128-d4-groups.json", + "sha256": "a397414bb0229fb1ebdb38798aa781fb89dbb9d5cdbed94c7cd2e9120da62718", + "max_group_size": 10, + "term_count": 75324, + "group_count": 7576, + "cell_norm_upper": [ + 6472926505087887788180917647763, + 1000000000000000000000000000000 + ] + }, + "steps": 97, + "group_exponentials": 2911, "contributions": { "degree4": [ - 5940239109672385481139, - 7242557440000000000000000000 + 58256338545790990093628258829867, + 110661601250000000000000000000000000000 ], "degree5": [ 5940239109672385481139, - 105017082880000000000000000000 + 42936701285000000000000000000 ], "degree6": [ 1841434385955714104548091490621189043370256643471762447183931887538056269601500787798709217, - 34109548519424000000000000000000000000000000000000000000000000000000000000000000000000000000000000 + 11661608069006000000000000000000000000000000000000000000000000000000000000000000000000000000000000 ], "degree7": [ 5357976462796315520172382471863567130110769930415287341551795662614168808804502363396127651, - 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"published_recursive_suzuki_audit": [ { @@ -124,9 +135,10 @@ ], "claims": { "global_twofold_target_met": true, + "global_fourfold_target_met": true, "exact_improvement_ratio": [ 11791, - 3481 + 2911 ] } } diff --git a/tracks/qcs/solutions/WangTheoPhys/issue128/certificates/issue128-d4-groups.json b/tracks/qcs/solutions/WangTheoPhys/issue128/certificates/issue128-d4-groups.json new file mode 100644 index 000000000..d4a1fc22d --- /dev/null +++ b/tracks/qcs/solutions/WangTheoPhys/issue128/certificates/issue128-d4-groups.json @@ -0,0 +1 @@ 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diff --git a/tracks/qcs/solutions/WangTheoPhys/issue128/scripts/build_v3_certificate.py b/tracks/qcs/solutions/WangTheoPhys/issue128/scripts/build_v3_certificate.py index 0840838b5..bb917fed0 100644 --- a/tracks/qcs/solutions/WangTheoPhys/issue128/scripts/build_v3_certificate.py +++ b/tracks/qcs/solutions/WangTheoPhys/issue128/scripts/build_v3_certificate.py @@ -1,13 +1,21 @@ #!/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 @@ -18,12 +26,72 @@ 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) @@ -43,6 +111,23 @@ def main() -> None: 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: @@ -52,6 +137,7 @@ def main() -> None: constants, n_sites, middle, + d4_site_override=d4_site_bound, ).global_error_bound <= tolerance ): @@ -63,11 +149,13 @@ def main() -> None: 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 @@ -120,11 +208,22 @@ def main() -> None: }, "candidate": { "formula": "five_copy_suzuki_fourth_order", - "proof_method": "local_log_E5_plus_E7_majorant_plus_exact_generator_tail", + "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": { @@ -155,12 +254,16 @@ def main() -> None: "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( diff --git a/tracks/qcs/solutions/WangTheoPhys/issue128/src/trottercert/verify.py b/tracks/qcs/solutions/WangTheoPhys/issue128/src/trottercert/verify.py index 45f5d42c3..ac1c125a8 100644 --- a/tracks/qcs/solutions/WangTheoPhys/issue128/src/trottercert/verify.py +++ b/tracks/qcs/solutions/WangTheoPhys/issue128/src/trottercert/verify.py @@ -1,31 +1,120 @@ 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 .intervals import cube_root_four_interval 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: @@ -239,6 +328,7 @@ def _verify_v3( data: dict[str, object], *, deep: bool, + certificate_path: Path, ) -> dict[str, object]: benchmark = data["benchmark"] if benchmark["normalization"] != EXPECTED_NORMALIZATION: @@ -275,9 +365,26 @@ def _verify_v3( if ( candidate["formula"] != "five_copy_suzuki_fourth_order" or candidate["proof_method"] - != "local_log_E5_plus_E7_majorant_plus_exact_generator_tail" + != ( + "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"]) @@ -295,6 +402,13 @@ def _verify_v3( ) 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 @@ -316,6 +430,7 @@ def _verify_v3( if deep: from .refined_error import ( build_refined_fourth_order_constants, + certified_d4_cell_coefficients, evaluate_refined_fourth_order_bound, ) from .rigorous_fourth import ( @@ -337,15 +452,23 @@ def _verify_v3( 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") @@ -359,6 +482,8 @@ def _verify_v3( 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", @@ -369,9 +494,16 @@ def _verify_v3( "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, } @@ -380,7 +512,8 @@ def verify_certificate( *, deep: bool = False, ) -> dict[str, object]: - data = json.loads(Path(path).read_text()) + certificate_path = Path(path) + data = json.loads(certificate_path.read_text()) schema = data.get("schema_version") if schema == 1: if deep: @@ -389,5 +522,9 @@ def verify_certificate( if schema == 2: return _verify_v2(data, deep=deep) if schema == 3: - return _verify_v3(data, deep=deep) + 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_resources_verify.py b/tracks/qcs/solutions/WangTheoPhys/issue128/tests/test_resources_verify.py index b9c670573..b81967352 100644 --- a/tracks/qcs/solutions/WangTheoPhys/issue128/tests/test_resources_verify.py +++ b/tracks/qcs/solutions/WangTheoPhys/issue128/tests/test_resources_verify.py @@ -1,7 +1,11 @@ +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, @@ -9,7 +13,7 @@ symmetric_group_exponentials, three_l_path_resources, ) -from trottercert.verify import verify_certificate +from trottercert.verify import _verify_d4_sidecar, verify_certificate def test_integer_roots_and_stage_merging() -> None: @@ -62,11 +66,94 @@ def test_verifier_recomputes_and_rejects_corruption(tmp_path) -> None: 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["candidate_steps"] == 116 + 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 From 4aa74f81a9256bb43dfcc0865beac0dc6db27130 Mon Sep 17 00:00:00 2001 From: JunkaiWang-TheoPhy <1181100960@qq.com> Date: Wed, 29 Jul 2026 21:56:59 +0800 Subject: [PATCH 09/27] document fourfold certified resource reduction --- tracks/qcs/solutions/WangTheoPhys/README.md | 71 ++++++++++++------- .../issue128-small-crosscheck.json | 8 +-- .../certificates/issue128-verification.txt | 42 ++++++----- .../issue128/src/trottercert/crosscheck.py | 8 +-- .../issue128/tests/test_small_crosscheck.py | 1 + 5 files changed, 79 insertions(+), 51 deletions(-) diff --git a/tracks/qcs/solutions/WangTheoPhys/README.md b/tracks/qcs/solutions/WangTheoPhys/README.md index 884b80f3b..4fe1cdf8b 100644 --- a/tracks/qcs/solutions/WangTheoPhys/README.md +++ b/tracks/qcs/solutions/WangTheoPhys/README.md @@ -18,26 +18,26 @@ 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 116 Trotter +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 | 116 | -| Merged group exponentials | 11,791 | 3,481 | -| Bond propagators | 848,952 | 250,632 | -| CNOT upper bound (3 per bond) | 2,546,856 | 751,896 | +| 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 / 3481 = 3.3872450445274347... +11791 / 2911 = 4.050498110614909... ``` -At 116 steps the certified global operator-norm error is -`9.72530161861392e-7`. At 115 steps the bound is -`1.0097845503628042e-6`, so 116 is the smallest integer accepted by this +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 @@ -56,10 +56,26 @@ Our proof instead: 3. maps those Lie elements into the concrete four-matching Heisenberg representation; 4. combines identical local Pauli strings before taking a norm bound; -5. converts the logarithm into the exact right generator; and -6. closes every degree at finite step size with a rational geometric locality +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`. @@ -93,20 +109,23 @@ 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 116 steps +## Error ledger at 97 steps | Right-generator degree | Global contribution | |---:|---:| -| 4 | `8.201852948883738e-7` | -| 5 | `5.6564503095749916e-8` | -| 6 | `5.398589151384083e-8` | -| 7 | `5.4166008370568104e-9` | -| 8 and above | `3.637787152637068e-8` | -| **Total** | **`9.72530161861392e-7`** | +| 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 @@ -135,12 +154,15 @@ their empirical constants cannot replace the pinned rigorous baseline. ## Independent checks and negative results -- A deep verifier regenerates the published baseline, local degree-five - coefficient, degree-seven majorant, finite tail, integer step boundary, and - resource arithmetic. +- 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 `4.463500595491358e-10`, below its outward-rounded certificate - `2.7027777777777777e-8`. + 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 @@ -175,8 +197,7 @@ PYTHONPATH=src python scripts/build_v3_certificate.py ``` The default test suite excludes explicitly marked research-reproduction tests. -The submitted run completed with 58 passing tests; the deep verifier also -completed successfully. +The deep verifier independently regenerates the expensive local algebra. ## Files diff --git a/tracks/qcs/solutions/WangTheoPhys/issue128/certificates/issue128-small-crosscheck.json b/tracks/qcs/solutions/WangTheoPhys/issue128/certificates/issue128-small-crosscheck.json index 9779f2e94..4405cc81b 100644 --- a/tracks/qcs/solutions/WangTheoPhys/issue128/certificates/issue128-small-crosscheck.json +++ b/tracks/qcs/solutions/WangTheoPhys/issue128/certificates/issue128-small-crosscheck.json @@ -1,10 +1,10 @@ { "length": 2, "n_sites": 4, - "model_note": "degenerate 2x2 periodic algebra sanity check for schema-v3 r=116", - "steps": 116, - "empirical_operator_norm_error": 4.463500595491358e-10, - "certified_upper_bound": 2.7027777777777777e-08, + "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-verification.txt b/tracks/qcs/solutions/WangTheoPhys/issue128/certificates/issue128-verification.txt index 66c1e47af..9a0eeb09c 100644 --- a/tracks/qcs/solutions/WangTheoPhys/issue128/certificates/issue128-verification.txt +++ b/tracks/qcs/solutions/WangTheoPhys/issue128/certificates/issue128-verification.txt @@ -6,7 +6,7 @@ Status valid: true verification_level: deep deep_proof_regenerated: true -test_suite: 58 passed, 7 slow tests deselected +test_suite: pending final full-suite rerun Benchmark --------- @@ -33,37 +33,43 @@ CNOT upper bound: 2546856 Candidate certificate --------------------- formula: the same five-copy fourth-order Suzuki formula -method: local log E5 + E7 majorant + exact right-generator tail +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=116: -degree-4 contribution: 8.201852948883738e-7 -degree-5 contribution: 5.6564503095749916e-8 -degree-6 contribution: 5.398589151384083e-8 -degree-7 contribution: 5.4166008370568104e-9 -degree >= 8 tail: 3.637787152637068e-8 -global error upper bound: 9.72530161861392e-7 +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=115: -global error upper bound: 1.0097845503628042e-6 +At r=96: +global error upper bound: 1.050565873970784e-6 -Therefore r=116 is the smallest step count accepted by this certificate. +Therefore r=97 is the smallest step count accepted by this certificate. Candidate resources ------------------- -steps: 116 -group exponentials: 3481 -bond propagators: 250632 -CNOT upper bound: 751896 +steps: 97 +group exponentials: 2911 +bond propagators: 209592 +CNOT upper bound: 628776 Certified comparison -------------------- -exact group/CNOT improvement ratio: 11791/3481 -decimal improvement: 3.3872450445274347 +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 ------------------------------------------- diff --git a/tracks/qcs/solutions/WangTheoPhys/issue128/src/trottercert/crosscheck.py b/tracks/qcs/solutions/WangTheoPhys/issue128/src/trottercert/crosscheck.py index 523e76659..6bd05747c 100644 --- a/tracks/qcs/solutions/WangTheoPhys/issue128/src/trottercert/crosscheck.py +++ b/tracks/qcs/solutions/WangTheoPhys/issue128/src/trottercert/crosscheck.py @@ -9,10 +9,10 @@ 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=116. +# 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(973, 10**9) -REFINED_STEPS = 116 +REFINED_BOUND_N144 = Fraction(996, 10**9) +REFINED_STEPS = 97 def small_exact_crosscheck( @@ -49,7 +49,7 @@ def small_exact_crosscheck( "length": length, "n_sites": n_sites, "model_note": ( - "degenerate 2x2 periodic algebra sanity check for schema-v3 r=116" + "degenerate 2x2 periodic algebra sanity check for schema-v3 r=97" if length == 2 else "periodic square lattice" ), diff --git a/tracks/qcs/solutions/WangTheoPhys/issue128/tests/test_small_crosscheck.py b/tracks/qcs/solutions/WangTheoPhys/issue128/tests/test_small_crosscheck.py index 879a0be4b..cfbfec7f9 100644 --- a/tracks/qcs/solutions/WangTheoPhys/issue128/tests/test_small_crosscheck.py +++ b/tracks/qcs/solutions/WangTheoPhys/issue128/tests/test_small_crosscheck.py @@ -7,3 +7,4 @@ 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 From 5b6361c6381dad31a072481e7c831891c670314a Mon Sep 17 00:00:00 2001 From: JunkaiWang-TheoPhy <1181100960@qq.com> Date: Wed, 29 Jul 2026 22:02:06 +0800 Subject: [PATCH 10/27] record final certificate verification --- .../issue128/certificates/issue128-verification.txt | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/tracks/qcs/solutions/WangTheoPhys/issue128/certificates/issue128-verification.txt b/tracks/qcs/solutions/WangTheoPhys/issue128/certificates/issue128-verification.txt index 9a0eeb09c..a88b1069b 100644 --- a/tracks/qcs/solutions/WangTheoPhys/issue128/certificates/issue128-verification.txt +++ b/tracks/qcs/solutions/WangTheoPhys/issue128/certificates/issue128-verification.txt @@ -6,7 +6,7 @@ Status valid: true verification_level: deep deep_proof_regenerated: true -test_suite: pending final full-suite rerun +test_suite: 70 passed, 9 slow tests deselected (Python 3.12) Benchmark --------- From c550a6b0915ccf7a1db410ee765ff65d99f96b6f Mon Sep 17 00:00:00 2001 From: JunkaiWang-TheoPhy <1181100960@qq.com> Date: Wed, 29 Jul 2026 22:05:15 +0800 Subject: [PATCH 11/27] audit fivefold and tenfold follow-up thresholds --- tracks/qcs/solutions/WangTheoPhys/README.md | 4 + .../docs/followups/2026-07-29-5x-10x-audit.md | 118 ++++++++++++++++++ 2 files changed, 122 insertions(+) create mode 100644 tracks/qcs/solutions/WangTheoPhys/issue128/docs/followups/2026-07-29-5x-10x-audit.md diff --git a/tracks/qcs/solutions/WangTheoPhys/README.md b/tracks/qcs/solutions/WangTheoPhys/README.md index 4fe1cdf8b..7c297fa87 100644 --- a/tracks/qcs/solutions/WangTheoPhys/README.md +++ b/tracks/qcs/solutions/WangTheoPhys/README.md @@ -170,6 +170,10 @@ their empirical constants cannot replace the pinned rigorous baseline. - 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. ## Reproduce 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..80c20962c --- /dev/null +++ b/tracks/qcs/solutions/WangTheoPhys/issue128/docs/followups/2026-07-29-5x-10x-audit.md @@ -0,0 +1,118 @@ +# Issue #128: certified 5× and 10× 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. From a5f618f17df44038facb06063fa131b7b8860ffc Mon Sep 17 00:00:00 2001 From: JunkaiWang-TheoPhy <1181100960@qq.com> Date: Thu, 30 Jul 2026 18:41:57 +0800 Subject: [PATCH 12/27] docs(issue128): design final delivery package --- .../2026-07-30-issue-128-delivery-design.md | 143 ++++++++++++++++++ 1 file changed, 143 insertions(+) create mode 100644 docs/plans/2026-07-30-issue-128-delivery-design.md 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. From dc724cbaa5c2780ef93812a063f0041244fe78b9 Mon Sep 17 00:00:00 2001 From: JunkaiWang-TheoPhy <1181100960@qq.com> Date: Thu, 30 Jul 2026 18:42:57 +0800 Subject: [PATCH 13/27] docs(issue128): plan final delivery package --- .../2026-07-30-issue128-final-delivery.md | 188 ++++++++++++++++++ 1 file changed, 188 insertions(+) create mode 100644 tracks/qcs/solutions/WangTheoPhys/issue128/docs/superpowers/plans/2026-07-30-issue128-final-delivery.md 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. From 4d38ef54690ac6ff0dbbc62f5d5ea3f2cdef230c Mon Sep 17 00:00:00 2001 From: JunkaiWang-TheoPhy <1181100960@qq.com> Date: Thu, 30 Jul 2026 19:17:33 +0800 Subject: [PATCH 14/27] Certify the exact degree-five follow-up Add streamed cubic-field algebra, exact same-support anticommuting groups, a canonical compressed sidecar, and regression tests for the D5 feasibility subproblem. Constraint: The D5 sidecar is supporting evidence and does not certify a 78-step global error bound Tested: pytest -q tests/test_delivery_package.py tests/test_cubic_field.py tests/test_support_groups.py; PYTHONPATH=src python scripts/build_d5_certificate.py --verify-only; shasum -a 256 certificates/issue128-d5-groups.json.gz Co-authored-by: OmX --- .../certificates/issue128-d5-groups.json.gz | Bin 0 -> 7838940 bytes .../issue128/scripts/build_d5_certificate.py | 131 ++++++++ .../issue128/src/trottercert/cubic_field.py | 163 ++++++++++ .../issue128/src/trottercert/cubic_local.py | 212 +++++++++++++ .../src/trottercert/support_groups.py | 282 ++++++++++++++++++ .../issue128/tests/test_cubic_field.py | 74 +++++ .../issue128/tests/test_cubic_local.py | 30 ++ .../issue128/tests/test_support_groups.py | 90 ++++++ 8 files changed, 982 insertions(+) create mode 100644 tracks/qcs/solutions/WangTheoPhys/issue128/certificates/issue128-d5-groups.json.gz create mode 100644 tracks/qcs/solutions/WangTheoPhys/issue128/scripts/build_d5_certificate.py create mode 100644 tracks/qcs/solutions/WangTheoPhys/issue128/src/trottercert/cubic_field.py create mode 100644 tracks/qcs/solutions/WangTheoPhys/issue128/src/trottercert/cubic_local.py create mode 100644 tracks/qcs/solutions/WangTheoPhys/issue128/src/trottercert/support_groups.py create mode 100644 tracks/qcs/solutions/WangTheoPhys/issue128/tests/test_cubic_field.py create mode 100644 tracks/qcs/solutions/WangTheoPhys/issue128/tests/test_cubic_local.py create mode 100644 tracks/qcs/solutions/WangTheoPhys/issue128/tests/test_support_groups.py 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 0000000000000000000000000000000000000000..2ffd6435eda5a3bae674395fe84cd2fb81333ba1 GIT binary patch literal 7838940 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