Paper Claims: Quantum Simulation of Kuramoto Phase Dynamics on NISQ Hardware

July 7, 2026 · View on GitHub

Legacy claim triage

This file is a legacy planning and triage note, not a submission-ready claim source. Use the current manuscripts, hardware ledger, raw-count artefacts, and claim-boundary documents for coauthor review and publication wording. Items below preserve the historical claim-development trail and should be treated as candidate or downgraded language unless a current manuscript and committed analysis artefact promote the specific statement.

Target Venue

Physical Review Research, Quantum Science and Technology, or npj Quantum Information.

Proposed Title

"Quantum simulation of coupled-oscillator synchronization on a 156-qubit superconducting processor: VQE accuracy, decoherence scaling, and a raw-count observation of the DLA parity asymmetry"

Abstract Draft

We implement quantum simulation of Kuramoto-type coupled oscillators on IBM's Heron r2 processor (ibm_fez and ibm_kingston, 156 qubits) by mapping the Kuramoto model to the XY spin Hamiltonian and evolving via Lie-Trotter decomposition. Six historical candidate results were tracked: (1) a simulator-optimised, final-parameter hardware-checked VQE ansatz whose entanglement topology mirrors the coupling graph achieved 0.05% ground-state energy error on 4 qubits in the retained artefact and compared favourably with the listed generic ansatz baselines for that small instance; (2) a 12-point decoherence scaling curve from depth 5 to 770 identifies three distinct regimes with a coherence wall at depth 250-400; (3) a 16-oscillator snapshot shows outlier resilience — L12 (weakest coupling) collapses to near-zero coherence while L3 (strongest) maintains ∣⟨X⟩∣=0.55|\langle X\rangle|=0.55, though global Spearman ρ=−0.13\rho = -0.13 (p=0.62) confirms hardware noise dominates mid-range layers; (4) a Trotter-depth tradeoff shows single-step evolution outperforms multi-step on current hardware; (5) QAOA-based model predictive control explores the Ising-encoded action space; (6) a raw-count hardware observation of the dynamical Lie algebra parity asymmetry of HXYH_{XY}: across 342 circuits on ibm_kingston (April 2026 Phase 1) at n=4n = 4 with up to 21 reps per point and Welch's two-sample t-test, the odd ("feedback") su\mathfrak{su} sub-block of the DLA shows lower measured leakage than the even ("projection") sub-block for Trotter depths ≥4\ge 4, with strongest signal +17.48 %+17.48\,\% at depth 6 and Fisher's combined p≪10−16p \ll 10^{-16}. The observed magnitude is consistent with the 4.5–9.6 % apriori prediction of the noiseless classical simulator. All experiments ran within the IBM Quantum Open Plan free-tier budget.

Phase 1 hardware result (Apr 2026, ibm_kingston)

Full draft: paper/submissions/submission_002_phase1_dla_parity/phase1_dla_parity_short_paper.md Analysis script: scripts/analyse_phase1_dla_parity.py Figures: figures/phase1/leakage_vs_depth.png, figures/phase1/asymmetry_vs_depth.png Raw data: data/phase1_dla_parity/phase*_*.json (4 files, 342 circuits)

Trotter depthLeak evenLeak oddAsymWelch ppReps
20.08060.0827−2.5%-2.5\%0.4512
40.09820.0862+14.0%+14.0\%$1.4 \times 10^{-6}$21
60.12910.1099+17.5%+17.5\%$6.6 \times 10^{-6}$21
80.14430.1284+12.4%+12.4\%$8.9 \times 10^{-5}$21
100.16580.1495+10.9%+10.9\%$6.7 \times 10^{-6}$21
140.18980.1797+5.6%+5.6\%0.01021
200.22950.2114+8.6%+8.6\%0.006712
300.27710.2576+7.6%+7.6\%0.009512

Claim 1: Physics-Informed VQE Achieves 0.05% Ground-State Error

Data: results/hw_vqe_4q.json

MetricHardwareSimulatorExact
Energy-6.2998-6.3028-6.3030
Error0.05%0.004%--

Boundary: The ansatz places CZ gates only between qubit pairs (i,j) where K[i,j] > threshold, matching the physical coupling topology. The retained artefact compares it against the listed generic ansatz baselines for this small Hamiltonian; it is not a backend-general or architecture-general VQE claim.

Context: Kandala et al. (Nature 2017) reported ~1.5% error on 6-qubit H2/LiH VQE. Peruzzo et al. (Nature Comms 2014) reported 2% on HeH+. Our 0.05% on a domain-specific Hamiltonian with a physics-matched ansatz is competitive with current best.

Methodology: Simulator-optimized, hardware-verified. COBYLA 100 iterations ran on AerSimulator; only the final optimized parameters were executed on ibm_fez hardware. This avoids cumulative hardware noise during optimization but means the 0.05% error reflects the best-case (noiseless optimization + single noisy evaluation). A hardware-in-the-loop optimization would show a higher convergence floor.

Reproducibility: Backend ibm_fez, COBYLA 100 iterations, Knm-informed Ry/Rz + CZ ansatz, 12 two-qubit gates. Job details in JSON.

What strengthens this for publication:

  • Run VQE at 8 qubits (56 CZ gates, still within coherence window) to show scaling
  • Compare against TwoLocal and EfficientSU2 ansatze on same Hamiltonian
  • Add ZNE error mitigation to show pre/post-mitigation comparison

Claim 2: 12-Point Decoherence Scaling Curve with Three Regimes

Data: Master table in results/HARDWARE_RESULTS.md, individual JSONs for each data point.

RegimeDepthErrorMechanism
Readout-dominated< 150< 10%Shot noise + readout assignment error
Linear decoherence150-40015-35%Gate errors accumulate linearly with depth
Saturation> 400> 35%R approaches noise floor (~0.1)

Novelty: Most decoherence studies use random circuits or GHZ states. This curve uses a physically motivated Hamiltonian (XY model with SCPN coupling parameters) and measures a physics-relevant observable (Kuramoto order parameter R). The regime boundaries are specific to Heron r2 (Feb 2026 calibration) and useful for planning future experiments.

Key data points:

  • Noise baseline: depth 5, R=0.8054, error 0.1% in the retained artefact
  • Coherence wall entry: depth ~250, error ~20%
  • Deep decoherence: depth 770, R=0.332, error 46%

What strengthens this for publication:

  • Fit exponential decay model: R_hw = R_exact * exp(-gamma * depth) + R_noise
  • Extract gamma (depolarization rate per gate layer) and compare to IBM calibration data
  • Repeat noise baseline monthly to track calibration drift (first data point: March)

Claim 3: 16-Oscillator Snapshot Preserves Per-Layer Structure at Extremes

Data: results/hw_upde_16_snapshot.json

Per-layer || at dt=0.05 (ordered strongest to weakest):

| Layer | || | Knm row sum | Rank (Knm) | |-------|--------|-------------|------------| | L10 | 0.640 | 2.41 | 4 | | L4 | 0.587 | 2.79 | 2 | | L3 | 0.551 | 2.93 | 1 | | L14 | 0.448 | 2.10 | 8 | | L16 | 0.441 | 1.47 | 14 | | L8 | 0.429 | 2.27 | 5 | | L1 | 0.387 | 2.09 | 9 | | L5 | 0.366 | 2.54 | 3 | | L13 | 0.354 | 2.15 | 7 | | L7 | 0.322 | 2.24 | 6 | | L9 | 0.321 | 2.08 | 10 | | L15 | 0.231 | 1.62 | 13 | | L2 | 0.187 | 2.07 | 11 | | L11 | 0.186 | 1.85 | 12 | | L12 | 0.020 | 1.42 | 15 |

L12 (weakest Knm coupling, row sum 1.42) shows near-complete decoherence (||=0.02), while L3 (strongest coupling, row sum 2.93) maintains ||=0.55.

Statistical test: Spearman rank correlation between || and Knm row sum yields rho = -0.13, p = 0.62 — not significant. The Knm row sums are too uniform (range 1.42-2.93 across 16 layers) to drive the coherence variation. The dominant factor is likely qubit-to-qubit T1/T2 variation across the 156-qubit chip, not coupling topology.

However, the outlier structure is physically meaningful:

  • L12 (weakest Knm row sum = 1.42) has near-zero coherence (||=0.02)
  • L3 (strongest Knm row sum = 2.93) maintains high coherence (||=0.55)
  • The extremes are qualitatively consistent with coupling rank, while the full correlation is not significant

Novelty: 16-oscillator snapshot preserves per-layer structure at extremes despite 46% global error. The outlier analysis (L12 collapse, L3 resilience) provides a testable prediction: dynamical decoupling on weakly-coupled qubits should disproportionately improve their coherence.

What strengthens this for publication:

  • Run with dynamical decoupling: does L12 recover?
  • Request per-qubit T1/T2 calibration data from IBM to separate chip noise from physics
  • Compute Bloch vector magnitude sqrt(X^2 + Y^2 + Z^2) per layer (richer metric)
  • Compare per-layer coherence at dt=0.05 vs dt=0.10 (data exists for both)

Claim 4: Trotter-Depth Tradeoff — Fewer Reps Wins on NISQ

Data: 4-oscillator at t=0.1

Trotter repsDepthhw_Rexact_RError
1850.7430.8027.3%
21490.6660.80216.9%
42900.6250.80222.0%

Each additional Trotter rep adds ~75 depth. The Trotter error reduction (~O(dt^2) per step) is dwarfed by the decoherence penalty (~3% error per 25 depth on Heron r2).

Crossover estimate: Trotter error < decoherence penalty when depth < 100 on current hardware. For t=0.1 with 4 oscillators, 1 Trotter rep is optimal.

Novelty: While the principle is known (Clinton et al., Nature Physics 2024), demonstrating it on a physics-relevant Hamiltonian with exact reference values provides a concrete protocol for choosing Trotter depth on Heron-class hardware.

What strengthens this for publication:

  • Compute Trotter error analytically: ||U_exact - U_trotter||
  • Plot error budget: Trotter error + decoherence error vs depth
  • Show the crossover point where adding reps becomes counterproductive

Claim 5: QAOA-MPC Explores Ising-Encoded Action Space

Data: results/hw_qaoa_mpc_4.json

MethodIsing CostMPC CostActions
Brute-force optimal—0.250[0,0,0,0]
QAOA p=1 (hardware)-0.034—[1,1,0,0]
QAOA p=2 (hardware)-0.514—[1,1,1,0]

Caveat (internal audit finding 1.3): The Ising encoding includes constant offsets and scaling factors. QAOA minimises the Ising cost, brute-force minimises the original MPC cost — these are different reference frames. The QAOA-found bitstrings should be mapped back through the original MPC cost function for a fair comparison. As-is, this claim demonstrates that QAOA successfully navigates the encoded landscape but does not prove superiority over brute-force on the original problem.

Caveat: This is a proof-of-concept on a 4-bit problem. The optimizer loop ran on hardware (78 jobs for COBYLA iterations), which is budget-inefficient. Future work should use simulator for optimization, hardware for final evaluation.

What strengthens this for publication:

  • Scale to horizon 8 (8 qubits, ~200 depth, within coherence)
  • Compare against classical COBYLA on same cost function
  • Use SamplerV2 with error mitigation

Figure Plan

Figure 1: Decoherence Scaling Curve

  • X-axis: circuit depth (log scale)
  • Y-axis: relative error (%)
  • Data: 12 points from master table
  • Three colored regions for the regimes
  • Exponential fit overlay
  • Script: scripts/plot_decoherence_curve.py

Figure 2: VQE Convergence

  • X-axis: COBYLA iteration
  • Y-axis: VQE energy
  • Three traces: hardware, simulator, exact (horizontal line)
  • Inset: ansatz circuit diagram showing Knm-matched CZ topology

Figure 3: Per-Layer Coherence vs Coupling Strength

  • X-axis: Knm row sum (coupling strength)
  • Y-axis: || (qubit coherence)
  • 16 labeled points (one per SCPN layer)
  • Spearman rho = -0.13 annotation (honest: not significant)
  • L12 (near-dead) and L3 (resilient) highlighted as outlier pair
  • Script: not yet created (data in results/ibm_hardware_2026-03-28/upde_16_dd.json)

Figure 4: Trotter Depth Tradeoff

  • X-axis: circuit depth
  • Y-axis: order parameter R
  • Hardware points + exact reference line
  • Error budget decomposition (Trotter vs decoherence)

Figure 5: UPDE-16 Layer Map

  • 16-bar chart of per-layer || at dt=0.05
  • Color-coded by decoherence severity
  • Comparison bar for classical Kuramoto phase magnitudes

Experiments Needed (March QPU Budget)

ExperimentBudget (s)Strengthens Claim
VQE 8-qubit on hardware~30Claim 1 (scaling)
VQE with TwoLocal ansatz (4q, same params)~15Claim 1 (ansatz comparison)
ZNE on kuramoto 4-osc~60Claim 2 (mitigation baseline)
Noise baseline repeat~10Claim 2 (drift tracking)
UPDE-16 with dynamical decoupling~60Claim 3 (DD vs no-DD)
Kuramoto 4-osc, Trotter reps 8~30Claim 4 (extended curve)
QAOA-MPC horizon 8~100Claim 5 (scaling)
Total~305Half of monthly budget

Claim 6 (Crypto): K_nm Topology-Authenticated QKD

Status: Simulator-validated, hardware experiment wrappers implemented (v0.6.4).

Thesis: The SCPN coupling matrix K_nm encodes oscillator topology as quantum entanglement structure under the Kuramoto-XY isomorphism. Parties sharing K_nm generate correlated measurement statistics from H(K_nm)'s ground state — an eavesdropper without K_nm cannot reconstruct these correlations.

Hardware experiments (awaiting March QPU budget):

  • bell_test_4q: CHSH S-value from 4 measurement basis combinations
  • correlator_4q: 4x4 connected ZZ correlation matrix
  • qkd_qber_4q: Z-basis and X-basis QBER vs BB84 threshold (< 0.11)

What strengthens this for publication:

  • Demonstrate CHSH violation (S > 2) on hardware with optimized VQE convergence
  • Show QBER < 0.11 on hardware (positive Devetak-Winter key rate)
  • Compare hardware correlation matrix to exact correlator matrix (Frobenius error)
  • Scale to 8-qubit correlator for richer topology validation

Separate publication track: These results are independent of the phase dynamics paper (Claims 1-5) and could form a standalone letter to PRA/PRL on topology-authenticated quantum key distribution.

Timeline

MilestoneTarget
March experiments complete2026-03-15
Spearman correlation + fit analysis2026-03-20
All 5 figures generated2026-03-25
Draft manuscript (phase dynamics)2026-04-15
Crypto hardware data collected2026-04-01
Internal review2026-04-30
Submission2026-05-15