SPDX-License-Identifier: AGPL-3.0-or-later¶
Commercial license available¶
© Concepts 1996–2026 Miroslav Šotek. All rights reserved.¶
© Code 2020–2026 Miroslav Šotek. All rights reserved.¶
ORCID: 0009-0009-3560-0851¶
Contact: www.anulum.li | protoscience@anulum.li¶
scpn-quantum-control — Results¶
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. |
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, S=2.188 (>8σ) | Legacy ibm_fez artefact row |
| 3 | QKD row | QBER 5.5% < BB84 threshold (11%) | 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 |
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.
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/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, p3.773718e-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 chi246.531552, p1.883218e-07, 2/4 significant depths.n=8: Fisher chi229.420107, p2.675193e-04, 3/4 significant depths.- IBM-reported usage:
305quantum seconds for jobibm-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_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/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.
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 (>8σ violation)
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.
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.
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 | 0.004 ms | 20.9 ms (Qiskit) | 5401× |
| Hamiltonian construction | 8 | 0.4 ms | 63 ms (Qiskit) | 158× |
| 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× |