[QCS] 🌠 Wander/漫步者: Issue #128 Certify a 4.050498× Tighter Trotter Resource Bound

[JunkaiWang-TheoPhy]

A future becomes trustworthy not when we prefer it, but when we can still bound its error.

一个未来值得信任，不因我们偏爱它，而因它的误差仍可被界定。

第二幕 · 让计算穿过物质的风暴

第七章 · 用一把误差尺裁定未来

工具离开工作台以后，第一场风暴来自误差。我们不凭一次漂亮运行判断未来，而让最坏情况也进入证书，

用一把可以复查的尺，把 2× 的目标推进到 4.050×。

← 上一幕：从答案铺回问题的路     ·     下一章：森林选择方向之前 →

Team

Team name	Wander (漫步者)
Members	Chenxi Wan, Yedi Shen, Junkai Wang
Contact email	WangTheoPhys@outlook.com

Challenge

Row
Challenge	Produce a machine-checkable, worst-case operator-norm Trotter certificate for a concrete two-dimensional Heisenberg Hamiltonian and reduce the provable compiled gate count by at least 2× below a pinned published rigorous bound.
Catalog issue	Closes #128 — “Tighter provable Trotter error bounds for a concrete 2D Hamiltonian.”
Track	qcs — Quantum Circuit Simulation.

Certified result

For the periodic spin-1/2 square-lattice isotropic Heisenberg benchmark

H = Σ_<i,j> (X_i X_j + Y_i Y_j + Z_i Z_j) / 4
L = 12, N = 144, T = 1, operator-norm tolerance = 1e-6

the submitted exact-rational certificate compares the same five-copy fourth-order Suzuki formula, four bond matchings, and compilation model on both sides.

Resource	Published rigorous control	Certified result	Reduction
Trotter steps	393	97	75.32%
Merged group exponentials	11,791	2,911	75.31%
Bond propagators	848,952	209,592	75.31%
CNOT upper bound	2,546,856	628,776	75.31%

The exact improvement is

11791 / 2911 = 4.050498110614909...

The challenge's required twofold target is therefore fully met: the result is 202.5% of the minimum ratio, or 102.5% beyond it. At 97 steps, the outward-rendered global error upper bound is 9.958938494314325e-7; at 96 steps it is 1.050565873970784e-6. The verifier decides this integer boundary by exact cross-multiplication.

Algorithmic and proof innovations

The resource gain comes entirely from a tighter certification architecture, not a changed circuit:

1. Compute the complete free-associative-word logarithm before applying norm inequalities.
2. Apply the Dynkin–Specht–Wever projection to recover degree-five and degree-seven Lie elements.
3. Evaluate the concrete four-matching Heisenberg representation, canonicalize translations, and combine identical Pauli strings before norming.
4. Certify an exact pairwise-anticommuting D4 partition: 75,324 canonical terms, 7,576 groups, maximum group size 10.
5. Reduce the D4 cell bound from 20.160968407335066 to 6.472926505087888, a factor 3.114660484942633.
6. Prove by exact enumeration that h=(XX+YY+ZZ)/4 has Pauli-l1 commutator growth constant 1 rather than the generic 3/2.
7. Close the finite-step right-generator ledger explicitly through D7 and bound D8-plus with a rational geometric locality tail.
8. Keep discovery outside the trusted path: the verifier uses integers, Fraction, outward rational intervals, and exact symplectic Pauli algebra.

Final evidence package

All links below are pinned to commit ebb055926302c7d911e5ab00805513c382ac458d:

• Six-page technical report PDF and LaTeX source
• Machine-readable JSON summary and plain-text summary
• Captured verification transcript, including stdout, stderr, and exit codes
• SHA-256 manifest binding the report, summaries, transcript, certificates, cross-check, and D4/D5 sidecars
• Reviewer-facing README

Frozen proof hashes:

main certificate  0a09623ce3b292a3637065c870fb3153bbdcddce30aef968565c4db3ddfc7201
D4 sidecar        a397414bb0229fb1ebdb38798aa781fb89dbb9d5cdbed94c7cd2e9120da62718
D5 sidecar        c5e8968a93b4497b41fe42c0e364324388272e183e1fcd20a536bc988f5361dd

Fivefold follow-up: evidence, not a claim

Fivefold resource arithmetic would require r <= 78; conditionally, G(78)=2341 and 11791/2341=5.036736437419906.... This PR now includes an independently checked D5 sidecar with 605,832 canonical Pauli terms, 123,106 same-support pairwise-anticommuting groups, and exact site-density upper bound

44948270001027856175670154896253 / 4000000000000000000000000000000
= 11.23706750025696...

The translation-coupled D4 budget and explicit D8-plus delayed-tail certificate remain unresolved hard gates. No 78-step global error certificate is claimed or supplied. The certified result of this PR is 4.050498×.

Verification completed on the pushed commit

• default solution suite: 89 passed, 10 deselected;
• delivery-focused suite: 19 passed;
• main verifier: valid: true, with 393/97 steps and exact ratio 11791/2911;
• independent D5 verifier: exact term count, group count, density, and fixed SHA-256 all pass;
• delivery self-check: delivery_check=PASS;
• manifest: all 10 entries report OK;
• PDF: 6 pages, all pages rendered and visually inspected; no undefined references, overfull boxes, or placeholder/tool-token text;
• git diff --check: clean.

Reproduce

cd tracks/qcs/solutions/WangTheoPhys/issue128
python -m pip install -e '.[test]'
pytest -q
PYTHONPATH=src python scripts/verify.py \
  certificates/issue128-certificate.json
PYTHONPATH=src python scripts/build_d5_certificate.py --verify-only
PYTHONPATH=src python scripts/package_delivery.py --check
shasum -a 256 -c artifacts/SHA256SUMS

The main numerical claim is stored as exact rational data; printed error upper bounds are rounded outward and never understate the certificate.

Long-form paper and hardened reviewer handoff

The original ten-file evidence package remains frozen at commit ebb055926302c7d911e5ab00805513c382ac458d. The following commit-pinned materials explain and extend that result without changing its certified claim:

• 30-page research manuscript PDF
• Modular LaTeX source and one-command build
• 23-reference citation audit
• Ten-minute presentation script and Q&A boundaries
• Short-report erratum

The erratum records that a legacy auxiliary recursive-Suzuki table in the frozen short report differs from the authoritative schema-v3 JSON. This does not affect the 393-versus-97 step boundary, the D4 certificate, or the exact 11791/2911 = 4.050498... resource ratio.

The follow-up also pins the RFC 1952 gzip OS header byte used by the frozen D5 sidecar. This restores byte-for-byte reproducibility on Python 3.14 while preserving the existing sidecar and SHA-256 c5e8968a93b4497b41fe42c0e364324388272e183e1fcd20a536bc988f5361dd.

Latest validation:

• 95 passed, 11 deselected;
• fast main verifier: valid=true;
• D5 verify-only: 605,832 terms, 123,106 groups, exact density passed;
• delivery self-check: delivery_check=PASS;
• all frozen manifest entries: OK;
• long manuscript: 30 pages, clean LaTeX log, visual audit complete;
• all 23 references validated by DOI, arXiv ID, or an official publisher catalog.

Claim boundary remains unchanged: no 78-step global error certificate is supplied, and no fivefold result is claimed.

[JunkaiWang-TheoPhy]

Final Issue 128 delivery is ready for review at ebb055926302c7d911e5ab00805513c382ac458d.

• certified result: 11791/2911 = 4.050498110614909...×;
• 97 steps accepted and 96 rejected by exact rational comparison;
• six-page PDF, JSON/TXT summaries, verifier transcript, D4/D5 sidecars, and one SHA-256 manifest are linked in the updated PR body;
• validation: 89 tests passed (10 slow tests deselected), delivery self-check passed, all 10 manifest entries verified;
• the D5 follow-up is supporting evidence only: no 78-step global error certificate is claimed or supplied.

The PR remains ready (not draft); maintainer review is requested.

[JunkaiWang-TheoPhy]

HPC sprint update: certified 4.1357418449× result and exact higher-order follow-up

This is a status update only; it does not claim a certified 5× result.

Certified result

The D5-integrated certificate now establishes the adjacent integer boundary:

• 95 steps: PASS
• 94 steps: FAIL

under the same 12×12 periodic Heisenberg instance, T=1, operator-norm tolerance 10⁻⁶, Suzuki formula, merging rule, and resource model as the baseline.

The resulting exact resource ratio is:

11791 / 2851 = 4.1357418449...

This supersedes the earlier certified 4.050498× result while leaving that earlier certificate valid. Local verification reports:

• fast verifier: valid=true
• deep regeneration: deep_proof_regenerated=true
• full regular regression: 110 passed, 11 deselected

D8 production outcome

The 31-stage exact D8 array 23044178 satisfied the requested production floor in measured scheduler wall time:

• 6 lighter stages completed;
• 25 heavy stages ran for approximately 4:30:11–4:30:13 and reached the wall-time limit;
• the afterok reducer was therefore cancelled automatically;
• no partial D8 sum was promoted to a global certificate.

The completed D8 stages remain useful cross-check evidence. In particular, the independent exact-D6 path agrees coefficient-by-coefficient with the D6 component extracted from completed D8 stages 25–30, including the full canonical Pauli-to-exact-coefficient map.

Current exact D6 route

To avoid the D8 serial-growth bottleneck, the follow-up computes E7 in independent free-word shards and reconstructs:

D6 = 7 E7 + (2/3) ad_A²(E5)

Multiple independent execution layouts are being used:

• 16-shard reference lane: 23044736
• 272-shard CPU fanout: 23044754
• 544-shard fine fanout: 23044779

Each lane writes hash-bound manifests and exact payloads. Reducers require complete interval coverage, matching source/payload hashes, and forward/reverse exact reductions before emitting a D6 sidecar. A separate comparison tool now verifies that independently reduced degree-6 payloads agree exactly and rejects modified payloads.

The finer fanout materially reduced the long-tail and per-process memory pressure. At the latest recorded snapshots:

• 272-way lane: 155/272 complete, 117 running, 0 failed after about five minutes;
• 544-way lane: 383/544 complete, 0 failed after about four minutes.

These are intermediate scheduler snapshots, not final scientific results.

Claim boundary

A conditional 78-step resource count would correspond to approximately 5.0367×, but 78 steps have not been certified. The multiplier will be updated again only if all of the following pass:

1. a complete exact D6 artifact is produced;
2. independent fanout reductions agree exactly;
3. the updated global error ledger passes at the new integer boundary;
4. fast and deep verification pass;
5. fresh-checkout reproduction passes.

Until then, the highest supported claim is the machine-verified 4.1357418449× reduction.

[JunkaiWang-TheoPhy]

Structural 5× audit: exact processor obstruction and full-phase 4×4 patch

This follow-up records two independently reproducible negative structural results. They narrow the next research direction, but they do not claim a certified 5× result and do not modify the current certified 4.1357418449× result.

1. Exact endpoint-processor obstruction

For the leading logarithmic defect (E_5), the exact Pauli/Hilbert–Schmidt overlap with the Heisenberg Hamiltonian is nonzero:

<H,E5> per 2×2 cell
= 3589/2400000
+ (30361/38400000) α
+ (2813/4800000) α²,
α³ = 4

∈ [0.004227229425845701..., 0.004227229425845703...]

For every endpoint conjugation generator (Q),

<H, i[Q,H]> = 0.

Therefore (E_5
otinmathrm{image}(mathrm{ad}_H)): an endpoint processor of the form (P(t)=exp(t^4Q)) cannot exactly cancel the full leading defect.

A separate support argument gives a stronger finite-range obstruction. The exact (E_5) ledger contains 74,448 nonzero Pauli terms:

Pauli support weight	Terms
2	144
4	8,448
6	65,856

Because commutation with the two-local (H) increases Pauli support by at most one, any processor with support at most four cannot reach the 65,856 weight-six terms. Those terms contain 29.52% of the squared Pauli-coefficient norm, implying a coefficient-(L_2) residual of at least 0.5433. This strictly rules out exact cancellation by the present three-site/six-color chirality basis and, more generally, by all support-(le4) processors.

The (L_2) residual is an algebraic diagnostic, not an operator-norm certificate. Longer-range processors may still partially reduce D4, but exact order lifting by endpoint conjugation is obstructed.

2. Full-phase open 4×4 patch audit

We also tested a strict open 4×4 construction that preserves the relative positions of all four colored-cell phases and leaves uncovered terms in the frozen anticommuting partition.

Quantity	Result
Complete D4 terms	75,324
Terms fitting the phase-preserving open 4×4 patch	2,562
Term coverage	3.40%
Pauli-(l_1) weight coverage	7.80%
Patch anticommuting bound	0.569928180
Frozen outside-term bound	6.169005017
Combined bound	6.738933197
Current certified D4 cell bound	6.472926505

The hybrid 4×4 construction is therefore 4.11% looser than the current certificate. The geometric reason is that 83.76% of D4 terms already have a bounding box larger than 4×4, with support extending to 8×9. Periodic 4×4 wrapping would alias distinct sites and is not a valid open-cluster certificate for the 12×12 operator.

This is a NO-GO for this specific open-4×4-plus-frozen-outside construction, not a universal lower bound against larger/overlapping patches, MPO/SOS, or translation-coupled local-marginal certificates.

3. Complete r=78 budget

Using the D5-integrated ledger, the current outward bounds at the 5× threshold (r=78) are:

Contribution	Bound
D4	(1.2590842 imes10^{-6})
Exact grouped D5	(9.3409537 imes10^{-8})
Current D6 majorant	(5.8406421 imes10^{-7})
D7	(8.7150631 imes10^{-8})
D8+ geometric tail	(9.0076388 imes10^{-7})
Total	(2.9244724 imes10^{-6})

Two statements are simultaneously true:

1. D4 alone exceeds the full tolerance, so D4 tightening is necessary.
2. Even with D4 set to zero, the remaining ledger is (1.6653883 imes10^{-6}), so D4 tightening alone is insufficient.

More strongly, even the idealized (D4=D6=0) ledger leaves

D5 + D7 + current tail = 1.0813240e-6 > 1e-6.

Thus a credible 5× route must jointly tighten D4, replace the generic D6 bound, and postpone or explicitly certify the high-degree tail.

There is also a compilation-cost constraint:

5× maximum groups = floor(11791/5) = 2358
bare r=78 groups = 30×78+1 = 2341
endpoint-processor allowance = 17 merged group exponentials

An algebraically useful processor that costs more than 17 additional merged exponentials would not preserve a strict 5× resource claim.

Consequence

The immediate exact-D6 work remains the highest-value short-horizon path and may move the certified result into the 4.18–4.27× range. A genuine 5× result, however, now requires a structure-changing certificate that simultaneously addresses D4 and the high-degree ledger; neither the present endpoint-processor family nor the strict open 4×4 patch provides that shortcut.