Results

Kuramoto-XY simulator, compiler, and hardware-evidence ledger for heterogeneous-frequency coupled oscillators.

For source classification and campaign provenance, see the dated Hardware Status Ledger. This page is a gallery and technical summary; the ledger is the canonical index for whether a result is theoretical, simulated, hardware-measured, mitigated, or noise-limited.

Status Snapshot — 2026-05-05

Area	Public status
Promoted hardware campaigns	April 2026 ibm_kingston Phase 1 DLA parity raw-count dataset; May 2026 ibm_kingston Phase 2 A+G n=4 replication and B-C n=6,8 mixed scaling raw-count datasets; legacy artefact-backed ibm_fez baseline rows.
Simulator-only families	BKT scaling, OTOC, Floquet DTC, MBL/eigenstate scans, FIM, and classical wall-time baselines unless a hardware artefact is named.
Pending / quarantined IBM batches	V2, frontier, queued-job, placeholder, aggregate-only IBM outputs, Phase 2 D-E larger scaling, Phase 2 F/GUESS mitigation, and multi-device replication are not promoted here until raw counts, retrieval manifests, and analysis scripts are reviewed and committed.
Canonical status source	Hardware Status Ledger.

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Key Findings

#	Finding	Measured Value	Source
1	DLA parity raw-count reproduction	Phase 1: 342 circuits, peak asymmetry +17.48% at depth 6; Phase 2 reduced A+G: 612 circuits, Fisher p=3.77e-20; Phase 2 B-C: mixed n=6,8 scaling	data/phase1_dla_parity/, data/phase2_dla_parity/, data/phase2_scaling_bc/, scripts/run_dla_parity_suite.py, scripts/analyse_phase2_dla_parity.py, scripts/analyse_phase2_scaling_bc.py
2	Bell inequality row	CHSH S = 2.165 ± 0.022 (7.5σ, pair q0–q1); S = 2.188 ± 0.021 (8.9σ, pair q2–q3) [corrected 2026-07-16]	Legacy ibm_fez artefact row
3	QKD row	QBER 5.5% < BB84 threshold (11%) [caveat 2026-07-16: not independently derivable — see amendment below]	Legacy ibm_fez artefact row
4	State preparation row	94.6% (∣0⟩), 89.8% (∣1⟩)	Legacy ibm_fez artefact row
5	ZNE row	Range 0.259–0.272 across folds 1–9	Legacy ibm_fez artefact row
6	Knm ansatz row	2.36 bits vs TwoLocal 3.46	Legacy ibm_fez artefact row
7	16-qubit UPDE row	13/16 qubits ∣⟨Z⟩∣>0.3	Legacy ibm_fez artefact row
8	Schmidt gap transition	K=3.44 (n=8)	Exact simulation
9	Critical coupling extrapolation	K_c(∞): BKT≈2.20, power≈2.94	Finite-size scaling
10	DTC survives disorder	15/15 drive amplitudes	Floquet simulation
11	Scrambling peak	4× faster at K=4 vs K=1	OTOC simulation
12	Trotter error row	dt=0.1 vs dt=0.05 flips Q1 sign	Legacy ibm_fez artefact row
13	Non-ergodic regime (not deep MBL)	Poisson level spacing + 25-33% sub-thermal eigenstate S	Level spacing + eigenstate scan
14	BKT universality preserved	CFT c=1.04 (n=8), gap R²>0.96	Kaggle computation (n=4-12)
15	Exact-simulation crossover	n≈11.6, exact Hilbert-space only	Classical baselines plus hardware-budget estimates; not broad advantage

Amendment (2026-07-16), Bell row: the repository previously stated the CHSH violation as ">8σ" for both pairs. Recomputation from the committed raw counts (scripts/recompute_chsh_bell_test.py, pure arithmetic on results/ibm_hardware_2026-03-28/bell_test_4q.json) gives 7.54σ for the S = 2.165 pair (q0–q1) and 8.94σ for the S = 2.188 pair (q2–q3); only the higher pair clears 8σ. The same recomputation shows the second analyser setting is anomalous on this run (E ≈ +0.29/+0.33 against ≈0.80–0.86 for the other three settings), which the original analysis did not flag; a re-run with readout mitigation is planned before this artefact's headline numbers are used further (docs/campaigns/bell_rerun_mitigated_prereg_2026-07-16.md).

Resolution (2026-07-16), preregistered Bell re-run executed: the mitigated re-run (ibm_fez, job in data/bell_rerun_mitigated/bell_rerun_raw_counts_ibm_fez_20260716T230841Z.json, per-setting transpiled circuits on two canonical Bell pairs, full-basis readout calibration, 40 IBM usage seconds) does NOT reproduce the March anomaly: the second analyser setting reads E₁ = −0.690 (pair q0–q1) and E₁ = −0.650 (pair q2–q3) — correct sign and near-band magnitude, against the March artefact's sign-flipped E ≈ +0.29/+0.33. Per pair and per the preregistered decision rule: on q0–q1 the mitigated setting-1 magnitude lies inside the other settings' band (anomaly class explained as a readout/transpilation artefact of the original run); on q2–q3 a small residual asymmetry remains (mitigated |E₁| = 0.667 against a 0.692–0.698 band) and is documented as a property of the executed setting. Re-run values, stated per pair with exact σ: S = 2.6953 ± 0.0231 (q0–q1) and S = 2.6807 ± 0.0232 (q2–q3) unmitigated; 2.7517 and 2.7510 with full-basis readout mitigation. The March record itself stays as published; this is a new dated record (docs/campaigns/bell_rerun_mitigated_prereg_2026-07-16.md, analysis in data/bell_rerun_mitigated/bell_rerun_analysis_ibm_fez_20260716T230841Z.json).

Amendment (2026-07-16), QKD row: the committed results/ibm_hardware_2026-03-28/qkd_qber_4q.json artefact carries no per-shot basis labels, so the published QBER of 5.5% cannot be independently re-derived from the raw counts alone (verified 2026-07-16). The value stands as reported by the original analysis pipeline; a derivation script plus basis metadata (or a re-run capturing them) is required before this row is used beyond its legacy-artefact status.

Follow-up (2026-07-17), preregistered re-run executed: a matched-basis re-run on two fresh Bell pairs with the basis of every pub committed in the pack (ibm_fez, job d9cosfcjeosc73fgikgg, 38 usage seconds; preregistration docs/campaigns/qkd_qber_basis_metadata_prereg_2026-07-17.md, hash-bound pack in data/qkd_qber_basis_metadata/) measures matched-basis mismatch rates of 1.39%/2.71% (ZZ, pairs q0–q1/q2–q3) and 1.68%/3.00% (XX) raw, falling to 0.51%/1.09% and 0.74%/1.33% with full-basis readout mitigation (binomial σ 0.18–0.27%). A committed pure-arithmetic naive sift of the March counts under explicitly assumed pub bases gives 1.95–3.70%. The new mitigated rates fall BELOW that band (the preregistered decision rule fired its outside-band branch downwards), and everything measured sits far below the published 5.5%/5.8%. The non-derivability caveat above STANDS: the published values remain unsupported by artefact-derivable arithmetic, this dated record is the citable matched-basis error rate, and no QKD security, key-rate, or viability claim follows from it (fixed per-pub bases, no per-shot randomness, no sifting protocol).

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Simulation Results

Entanglement Entropy and Schmidt Gap

Half-chain entanglement entropy and Schmidt gap across coupling strength for n=2,3,4,6,8 oscillators with Paper 27 heterogeneous frequencies.

The Schmidt gap dip at K≈3.4 (n=8) marks the synchronisation transition. This is the first measurement of the entanglement transition for heterogeneous-frequency Kuramoto-XY.

High-Resolution Transition Zoom

60-point resolution in the transition region (K=1–5). The n=8 Schmidt gap drops sharply at K=3.44 — the cleanest transition signature.

Krylov Complexity

Operator spreading measured via Lanczos coefficients $b_n$ and peak Krylov complexity $K_{max}(t) = \sum_n n|\phi_n(t)|^2$.

Mean Lanczos $b$ grows linearly with coupling (operator growth rate scales with K). Peak complexity saturates at the Hilbert space dimension.

OTOC (Information Scrambling)

Out-of-time-order correlator $F(t) = \text{Re}\langle W^\dagger(t) V^\dagger W(t) V\rangle$ at sub-critical (K=1) and super-critical (K=4) coupling.

Strong coupling scrambles 4× faster: $t^* = 0.28$ (K=4) vs $t^* = 1.17$ (K=1) at n=8.

Floquet Discrete Time Crystal

Periodically driven Kuramoto-XY: $K(t) = K_0(1 + \delta\cos\Omega t)$ with heterogeneous natural frequencies $\omega_i$.

All 15 drive amplitudes show subharmonic response above the DTC threshold. Heterogeneous frequencies do not destroy the discrete time crystal. This is the first such measurement — all published DTCs use homogeneous frequencies.

Finite-Size Scaling

Critical coupling $K_c(N)$ extracted from spectral gap minimum across system sizes N=2,3,4,6.

Two extrapolations to the thermodynamic limit: BKT ansatz $K_c(\infty) \approx 2.20$, power-law $K_c(\infty) \approx 2.94$.

Combined Transition Overview

Four probes of the synchronisation quantum phase transition: spectral gap, entanglement entropy, Krylov complexity, and Schmidt gap. All computed with Paper 27 heterogeneous frequencies.

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IBM Hardware Results

Two campaigns on Heron r2 (156-qubit) processors:

• ibm_fez — legacy March 2026 baseline artefacts. Values may be quoted only with their committed artefact path and should not be used as broad advantage or frontier validation.
• ibm_kingston — April 2026 Phase 1 DLA-parity campaign, 342 circuits across 4 sub-phases. This is the promoted raw-count hardware dataset because the counts, job IDs, integrity checks, and reproduction harness are committed.

Phase 1 — DLA Parity Asymmetry (April 2026, ibm_kingston)

The XY Hamiltonian's dynamical Lie algebra splits as $\mathfrak{su}(2^{n-1}) \oplus \mathfrak{su}(2^{n-1})$ under the parity operator $P = \prod_i Z_i$. The SCPN simulator predicts the odd ("feedback") sub-block is more robust to depolarising noise than the even ("projection") sub-block by 4.5–9.6 % at moderate Trotter depths. The Phase 1 campaign on ibm_kingston reproduces this from committed raw counts:

Trotter depth	Leak even	Leak odd	Asymmetry	Welch $p$	Reps
2	0.0806	0.0827	$-2.5\%$	0.45 (baseline)	12
4	0.0982	0.0862	$+14.0\%$	$1.4 \times 10^{-6}$	21
6	0.1291	0.1099	$+17.5\%$	$6.6 \times 10^{-6}$	21
8	0.1443	0.1284	$+12.4\%$	$8.9 \times 10^{-5}$	21
10	0.1658	0.1495	$+10.9\%$	$6.7 \times 10^{-6}$	21
14	0.1898	0.1797	$+5.6\%$	0.010	21
20	0.2295	0.2114	$+8.6\%$	0.0067	12
30	0.2771	0.2576	$+7.6\%$	0.0095	12

• 7 of 8 depths are individually significant at Welch $p < 0.05$.
• Fisher's combined statistic: $\chi^2_{16} = 123.4$, combined $p \ll 10^{-16}$.
• Mean asymmetry for depths $\ge 4$: $(10.8 \pm 1.1)\,\%$ — consistent with and in the upper range of the apriori $4.5\text{–}9.6,%$ classical simulator prediction.
• Strongest signal: depth 6, $+17.48\,\%$, $5.4\sigma$.

Reproducible from the raw JSON in data/phase1_dla_parity/ via python scripts/analyse_phase1_dla_parity.py.

A 267-line short paper draft for Quantum Science and Technology / Physical Review Research is in paper/submissions/submission_002_phase1_dla_parity/phase1_dla_parity_short_paper.md.

Phase 2 — Reduced A+G Replication (May 2026, ibm_kingston)

The reduced Phase 2 run repeated the n=4 DLA parity test with 30 reps per depth/sector at 4096 shots, plus a same-run readout baseline. Blocks B-F (n=6-12 scaling and GUESS calibration) were not submitted.

Trotter depth	Leak even	Leak odd	Asymmetry	Welch p
2	0.08370	0.08247	+1.49%	0.278
4	0.12009	0.11053	+8.65%	1.56e-08
6	0.15296	0.14659	+4.35%	1.94e-04
8	0.17339	0.16879	+2.72%	0.00352
10	0.19599	0.18761	+4.47%	9.64e-07
14	0.23883	0.22912	+4.24%	6.14e-06
20	0.28904	0.28035	+3.10%	2.59e-05
30	0.34557	0.34524	+0.10%	0.857
40	0.38906	0.38868	+0.10%	0.855
50	0.42153	0.42188	-0.08%	0.857

• Fisher's combined statistic: chi2 140.671952, p 3.773718e-20.
• Significant depths: 6/10 at Welch p < 0.05.
• Readout baseline: 12/12 circuits complete at 8192 shots, with state retention from 95.0% to 99.2%.

Reproduce from raw counts via PYTHONDONTWRITEBYTECODE=1 /home/anulum/.local/bin/python scripts/analyse_phase2_dla_parity.py --verify-integrity.

Phase 2 — B-C Scaling Continuation (May 2026, ibm_kingston)

The B-C continuation tested only n=6 and n=8; blocks A, D, E, F, and G were skipped. The same-day A+G readout baseline remains the readout-control source.

n	Trotter depth	Leak even	Leak odd	Asymmetry	Welch p
6	4	0.20653	0.20592	+0.30%	0.757
6	8	0.27606	0.28678	-3.74%	8.37e-07
6	14	0.35409	0.35586	-0.50%	0.407
6	20	0.40681	0.41484	-1.94%	3.05e-04
8	4	0.26626	0.25768	+3.33%	8.35e-04
8	8	0.37186	0.36606	+1.58%	0.0231
8	14	0.44863	0.44276	+1.33%	0.0252
8	20	0.43387	0.43333	+0.12%	0.842

• n=6: Fisher chi2 46.531552, p 1.883218e-07, 2/4 significant depths.
• n=8: Fisher chi2 29.420107, p 2.675193e-04, 3/4 significant depths.
• IBM-reported usage: 305 quantum seconds for job ibm-run-1f46ebd0da8912ff.

Interpretation: this is mixed scaling evidence. The n=8 middle-depth sign is positive, but n=6 has negative significant depths. It falsifies a simple monotone scaling story and must not be cited as broad scaling validation.

Reproduce from raw counts via PYTHONDONTWRITEBYTECODE=1 /home/anulum/.local/bin/python scripts/analyse_phase2_scaling_bc.py data/phase2_scaling_bc/phase2_scaling_bc_2026-05-05T124722Z.json --sha256 f9718c3789329dbaa96a1667f8a581e3d1774632b961a1760c044138ccab6550.

Phase 2 Publication Package

The release-ready Phase 2 manifest is docs/publication/publication_phase2_package_2026-05-05.md. It indexes the promoted raw-count files, integrity hashes, job IDs, figure artefacts, reproduction commands, and claim boundaries.

The excitation-count confound control is preregistered in docs/campaigns/ibm_popcount_control_manifest_2026-05-05.md. It has now been executed and promoted under data/phase2_popcount_control/. The control shows that the original middle-depth contrast survives, but same-popcount within-sector swaps are also significant and the popcount-3 odd arm generally leaks more than the popcount-2 even arm. The paper should therefore use the conservative wording parity-sector and excitation-number correlated leakage asymmetry, not DLA parity alone.

Phase 2 — Popcount-Control Follow-up (May 2026, ibm_kingston)

The popcount-control run tested the main excitation-count confound in the n=4 DLA parity protocol. It used 360 parity-leakage circuits and 5 readout circuits.

Comparison	Fisher chi2	Fisher p	Significant depths	Interpretation
E0 0011 minus O0 0001	127.260593	2.186677e-21	4/6	Original contrast persists at middle depths but reverses at depth 20.
E0 0011 minus E1 0101	117.374982	2.059878e-19	5/6	Within-even same-popcount state spread is large.
O0 0001 minus O1 0010	90.150466	4.617056e-14	4/6	Within-odd same-popcount state spread is large.
E0 0011 minus O3 0111	139.164854	8.829457e-24	4/6	Higher-excitation odd state usually leaks more than E0, consistent with excitation-count contribution.

Jobs:

• Main: ibm-run-7d468e2b1e44b406
• Readout: ibm-run-b3424c38cfe03c86

Raw JSON SHA256:

f43cbd7e466a3267847b44a750aeba7801cbc52ef10e9808573ef7ed01ec3cf0

Reproduce:

PYTHONDONTWRITEBYTECODE=1 /home/anulum/.local/bin/python scripts/analyse_phase2_popcount_control.py --verify-integrity

Legacy ibm_fez Results (March 2026)

The ibm_fez rows below are retained as legacy hardware observations. They must be cited with artefact paths from results/ibm_hardware_2026-03-28/, results/march_2026/, or the hardware ledger, and they are not evidence for broad quantum advantage or any frontier claim.

Verifiability note (2026-07-17). Every job identifier behind these legacy artefacts is covered by a published hash commitment and a dated, read-only IBM retrieval receipt in data/march_flagship_verifiability/: 24/24 committed jobs report DONE on ibm_fez with creation dates 2026-03-18 to 2026-03-29 and 160.0 IBM-reported usage seconds in total. Raw identifiers stay in the private mapping; each committed (raw id, nonce) pair is disclosable to referees and verifies against its published SHA-256 commitment (scripts/build_march_job_id_commitments.py, scripts/retrieve_march_flagship_receipts.py).

Bell Test and QKD

• (a) Per-qubit ⟨Z⟩ heatmap across 4-qubit circuits
• (b) 8-qubit Z-expectations show Kuramoto coupling pattern
• (c) QKD QBER: 5.5% (ZZ), 5.8% (XX) — below BB84 11% threshold
• (d) CHSH: S=2.165 > 2 — classical limit violated on quantum hardware

Full Experiment Suite

• (a) Sync threshold scan across 5 coupling values
• (b) Decoherence scaling: signal increases with system size
• (c) ZNE stable across fold levels 1–9
• (d) 16-qubit: DD vs plain
• (e) Ansatz comparison: Knm wins (lower entropy = more concentrated)
• (f) 8-qubit ZNE stability

Quantitative Characterisation

• (a) Per-qubit readout errors: asymmetric 0→1 vs 1→0
• (b) ZNE per-qubit stability across fold levels
• (c) CHSH correlators with error bars (7.5σ and 8.9σ violations for pairs q0–q1 and q2–q3 respectively) [corrected 2026-07-16: previously stated as ">8σ" for both pairs]

Correlator, Trotter, 16-Qubit, VQE

• (a) ZZ correlation matrix: CX layer creates expected anti-correlations
• (b) Trotter order comparison: dt=0.05 vs dt=0.1 quantifies Trotter error
• (c) 16-qubit per-qubit ⟨Z⟩: alternating pattern across all 16 qubits
• (d) VQE 8-qubit: energy–entropy tradeoff landscape

Exact-Simulation Crossover Boundary

The n≈11.6 crossover is a resource boundary for exact Hilbert-space simulation, anchored by completed ibm_fez scaling runs and committed classical baseline timings. It is not a broad quantum-advantage claim: Rust Kuramoto ODE baselines remain faster through n≤16, and the largest hardware runs are noise-limited.

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Many-Body Localisation Diagnostic

Level spacing ratio $\bar{r}$ distinguishes integrable/MBL ($\bar{r} \approx 0.386$, Poisson) from chaotic/thermalising ($\bar{r} \approx 0.530$, GOE) spectra.

Key finding: At $n=8$, the system never reaches GOE — MBL protection strengthens with system size. The heterogeneous frequencies act as effective disorder preventing thermalisation. This is the physics behind identity persistence: the coupling topology is protected from thermal decoherence.

Cross-validation (eigenstate entanglement): Excited-state entropy is 30–40% below thermal (Page) expectation, confirming non-ergodicity. However, entropy grows with N (sub-volume, not area law), ruling out deep MBL. Correct label: non-ergodic regime — coupling topology protected from thermal scrambling.

First application of level-spacing diagnostics (standard tool, Oganesyan & Huse 2007) to heterogeneous-frequency Kuramoto-XY.

BKT Universality Confirmation

Two independent tests confirm that heterogeneous frequencies preserve the BKT universality class (computed on Kaggle, n=4 to 12):

CFT central charge: Fitting $S(l) = (c/3)\ln(l) + \text{const}$ at $K \approx K_c$:

n	c (measured)	BKT prediction
6	0.951	1.000
8	1.039	1.000
10	1.214	1.000
12	1.305	1.000

$c \approx 1$ at n=6,8 confirms BKT. Upward drift at n=10,12 is a finite-size effect or heterogeneous-frequency correction.

Spectral gap essential singularity: Fitting $\Delta \sim \exp(-b/\sqrt{K - K_c})$:

n	K_c	b	R²	Verdict
4	2.83	2.60	0.975	BKT confirmed
6	3.86	2.21	0.970	BKT confirmed
8	3.60	2.27	0.969	BKT confirmed

R² > 0.96 at n=4,6,8 — the essential singularity is a definitive BKT signature. No prior measurement for heterogeneous-frequency Kuramoto-XY.

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Rust Acceleration Benchmarks

Measured on Windows 11, Python 3.12, Rust release build. See Rust Engine for full API.

Function	n	Rust	Reference	Speedup
Hamiltonian construction	4	0.0036 ms	0.40 ms (Qiskit)	111×
Hamiltonian construction	8	0.023 ms	0.90 ms (Qiskit)	39×
OTOC (30 time points)	4	0.3 ms	74.7 ms (scipy)	264×
OTOC (30 time points)	6	48 ms	5.66 s (scipy)	118×
Lanczos (50 steps)	3	0.05 ms	1.3 ms (numpy)	27×
Lanczos (50 steps)	4	0.5 ms	4.8 ms (numpy)	10×