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Preprint Outline

Title: Quantum simulation of coupled Kuramoto oscillators on IBM Heron r2: BKT transition, decoherence budget, and K_nm-informed ansatz advantage

Target venue: Physical Review Research (or Quantum Science and Technology)

Authors: Miroslav Šotek (ORCID 0009-0009-3560-0851)

Abstract (150 words)

We present a quantum simulation framework for the Kuramoto model of coupled oscillators, mapped to the XY Hamiltonian on IBM Heron r2 superconducting hardware. Using a coupling-topology-informed variational ansatz, we demonstrate 6× faster convergence to the ground state compared to hardware-efficient alternatives. We characterize the decoherence budget at 16 qubits, showing that the XY Hamiltonian's spectral structure amplifies T2-limited fidelity loss at specific Trotter depths. We connect the synchronization transition to the Berezinskii-Kosterlitz-Thouless (BKT) universality class and show that the persistent homology threshold p_h1 = 0.72, previously empirical, is within 0.5% of the product of BKT universal constants A_HP × sqrt(2/π). We provide an open-source package (1300+ tests) with ADAPT-VQE, VarQITE, QSVT resource estimation, and benchmarks against five physical coupled-oscillator systems.

I. Introduction

  • Coupled oscillators: Kuramoto model, synchronization, applications (power grids, neural networks, photosynthesis, plasma)
  • Quantum simulation on NISQ hardware: state of the art, IBM utility experiments (Nature 2023)
  • The Kuramoto-XY mapping: dθ/dt = ω + KΣsin(Δθ) → H = -Σ K(XX+YY) - Σ ωZ
  • Our contribution: physics-informed ansatz, BKT analysis, open-source framework

II. Methods

A. Hamiltonian construction

  • K_nm coupling matrix (Paper 27 definition)
  • XY Hamiltonian compilation to Qiskit SparsePauliOp
  • Trotter decomposition with commutator error bounds (analytical: 4Σ|K_ij||ω_j - ω_i|)

B. K_nm-informed ansatz

  • Standard approach: EfficientSU2 (generic entanglement pattern)
  • Our approach: CZ gates placed only where K_ij > threshold
  • Circuit depth reduction: O(n²) → O(nnz(K))

C. Hardware execution

  • IBM Heron r2 (ibm_fez): 156 qubits, CZ error ~0.5%, T2 ~100μs
  • PEA error mitigation (resilience_level=2)
  • Fractional RZZ gates (native on Heron, 50-68% depth reduction)

D. Analysis framework

  • BKT observables: Fiedler eigenvalue, vortex density, Wilson loops
  • Entanglement entropy scaling and CFT central charge
  • Quantum Fisher Information for parameter estimation

III. Results

A. Ansatz advantage (HARDWARE DATA)

  • Figure 1: VQE convergence — K_nm ansatz vs EfficientSU2 (6× speedup)
  • 12 hardware data points on ibm_fez (February-March 2026)
  • Energy relative error vs circuit depth

B. Decoherence budget

  • Figure 2: Fidelity vs Trotter depth at 4, 8, 16 qubits
  • Coherence budget: max useful depth = T2 / (2 × t_gate × n_2q)
  • Hardware data: noise baseline R = 0.784 (March) vs 0.805 (February)

C. BKT analysis

  • Figure 3: Phase diagram K_c vs T_eff
  • Figure 4: Entanglement entropy S(n/2) vs coupling K
  • Figure 5: Vortex density across synchronization transition
  • Finding: p_h1 = A_HP × sqrt(2/π) = 0.717 (0.5% from 0.72)
    • Caveat: A_HP measured on square lattice, not K_nm graph
    • Significance: if confirmed, consciousness gate threshold = BKT universal

D. Algorithm comparison

  • Table 1: QSVT (O(αt)) vs Trotter-1 (O((αt)²/ε)) vs Trotter-2
  • ADAPT-VQE: gradient-driven operator selection
  • VarQITE: guaranteed convergence without optimizer

E. Physical system benchmarks

  • Table 2: Topology correlation ρ for FMO, IEEE 5-bus, JJA, EEG, ITER
  • Finding: moderate correlations (0.2-0.4) — exponential decay is generic

IV. Resource estimation

  • Surface code: d=7 at p=0.3% → 1552 physical qubits for 16 oscillators
  • Circuit cutting: 32 oscillators via 2×16 partitions on Heron
  • GPU baseline: A100 beats QPU until N33 (statevector) or N25 (MPS at criticality)
  • Honest quantum advantage boundary: N > 40 for generic dynamics

V. Discussion

What we claim:

  • K_nm-informed ansatz provides measurable improvement (hardware-verified)
  • BKT framework correctly describes the XY model synchronization transition
  • p_h1 ≈ A_HP × sqrt(2/π) connects empirical threshold to BKT universals

What we do NOT claim:

  • Quantum advantage at 16 qubits (classical is faster)
  • The K_nm values model any specific physical system
  • "Consciousness" — we measure physical observables, not philosophical concepts
  • p_h1 derivation is exact (square lattice A_HP, not graph A_HP)

Reframe for reviewers:

  • This is a NISQ benchmarking study, not a quantum advantage claim
  • The K_nm ansatz is a general technique for structured Hamiltonians
  • The BKT connection is standard XY physics, applied to a novel graph
  • The consciousness interpretation is outside the scope of this paper

VI. Conclusion

Open-source package: github.com/anulum/scpn-quantum-control (AGPL-3.0) 1300+ tests, Rust acceleration, 5 physical system benchmarks, 4 simulation algorithms.

Data availability

Zenodo DOI: [to be created with v1.0.0 release] IBM hardware data: 12 data points on ibm_fez (February-March 2026)

References (~50 citations)

Key: Bastidas 2025, Babbush 2023, IBM Nature 2023, Kosterlitz-Thouless 1973, Hasenbusch-Pinn 1997, Calabrese-Cardy 2004, Grimsley ADAPT-VQE 2019, McArdle VarQITE 2019, Gilyén QSVT 2019, Huang shadows 2020