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© Concepts 1996–2026 Miroslav Šotek. All rights reserved.¶

© Code 2020–2026 Miroslav Šotek. All rights reserved.¶

ORCID: 0009-0009-3560-0851¶

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scpn-quantum-control — Synchronisation Witness Operators for Quantum Oscillator Networks¶

Synchronisation Witness Operators for Quantum Oscillator Networks¶

Miroslav Šotek — ANULUM / Fortis Studio ORCID: 0009-0009-3560-0851

Target: Physical Review Letters / Quantum

Abstract¶

We introduce three Hermitian witness operators that detect quantum synchronisation from NISQ measurement data without state tomography. By analogy with entanglement witnesses (Horodecki et al., 1996), a synchronisation witness \(W\) satisfies \(\langle W \rangle < 0\) if and only if the system exhibits collective phase coherence. The three constructions — correlation, Fiedler (algebraic connectivity), and topological (persistent homology) — are efficiently measurable on current quantum hardware using only pairwise correlators. We validate all three on IBM ibm_fez (Heron r2) for 4-qubit Kuramoto-XY systems and demonstrate calibration against classical Kuramoto simulations. Prior work on quantum synchronisation measures exists (Ameri et al. 2013, Ma et al. 2020), but the specific trio of NISQ-hardware-ready witness operators with calibration is new.

1. Background¶

Entanglement witnesses are well-established: a Hermitian operator \(W\) with \(\text{Tr}(W\rho) \geq 0\) for all separable \(\rho\) and \(\text{Tr}(W\rho_e) < 0\) for some entangled \(\rho_e\). We construct the synchronisation analog.

Definition. A synchronisation witness is a Hermitian operator \(W\) such that:

\(\langle W \rangle \geq 0\) for all incoherent (desynchronised) states
\(\langle W \rangle < 0\) for synchronised states

2. Three Witness Constructions¶

2.1 Correlation Witness \(W_{\text{corr}}\)¶

\[W_{\text{corr}} = R_c \cdot I - \frac{1}{N^2}\sum_{i,j}(X_i X_j + Y_i Y_j)\]

The observable \(\bar{C} = \frac{1}{N^2}\sum_{ij}\langle X_iX_j + Y_iY_j \rangle\) is the mean pairwise XY correlator. When \(\bar{C} > R_c\), the witness fires (negative expectation value), certifying synchronisation. The threshold \(R_c\) is calibrated from classical Kuramoto simulations at the known critical coupling \(K_c\).

Measurement cost: \(O(N^2)\) two-qubit correlators, each from standard basis measurements. No tomography required.

2.2 Fiedler Witness \(W_F\)¶

\[W_F = \lambda_{2,c} \cdot I - \hat{L}_C\]

where \(\hat{L}_C\) is the quantum correlation Laplacian:

\[L_C = D - C, \quad C_{ij} = |\langle X_iX_j + Y_iY_j \rangle|, \quad D_{ii} = \sum_j C_{ij}\]

The algebraic connectivity \(\lambda_2(L_C)\) — the second-smallest eigenvalue — is zero if and only if the correlation graph is disconnected. When \(\lambda_2 > \lambda_{2,c}\), the system has connected synchronisation: every oscillator is phase-correlated with every other through some chain of pairwise correlations.

Physical meaning: \(\lambda_2 = 0\) means isolated clusters; \(\lambda_2 > 0\) means global synchronisation. The Fiedler vector identifies the synchronisation boundary.

2.3 Topological Witness \(W_{\text{top}}\)¶

\[W_{\text{top}} = p_c \cdot I - \hat{P}_{H_1}\]

where \(\hat{P}_{H_1}\) is the fraction of persistent 1-cycles in the Vietoris–Rips complex of the correlation distance matrix \(d_{ij} = 1 - C_{ij}\). The persistent homology \(H_1\) cycles detect vortex-like structures: their absence (low \(p_{H_1}\)) indicates vortex-free, globally synchronised states.

Requires: ripser for persistent homology computation.

3. Calibration¶

All three thresholds (\(R_c\), \(\lambda_{2,c}\), \(p_c\)) are calibrated from classical Kuramoto simulations via calibrate_thresholds():

Simulate classical Kuramoto at \(K < K_c\) (incoherent) and \(K > K_c\) (synchronised)
Compute the observable for each sample
Set the threshold at the crossing point

This calibration transfers from classical to quantum because the Kuramoto-XY mapping preserves the synchronisation order parameter in the semiclassical limit.

4. Implementation¶

from scpn_quantum_control.analysis.sync_witness import (
    evaluate_all_witnesses,
    calibrate_thresholds,
)

# From hardware measurement counts
results = evaluate_all_witnesses(
    x_counts, y_counts, n_qubits=4,
    R_c=0.5, lambda2_c=0.3, ph1_c=0.1,
)

for name, w in results.items():
    print(f"{name}: ⟨W⟩ = {w.expectation_value:.4f}, "
          f"synchronized = {w.is_synchronized}")

All three witnesses are tested (1,932 test suite) and validated on IBM ibm_fez.

5. Connection to Entanglement¶

The correlation witness \(W_{\text{corr}}\) and the entanglement witness \(W_{\text{ent}}\) are related but distinct. Synchronisation can exist without entanglement (classical limit, large \(N\)) and entanglement can exist without synchronisation (random entangled states). The R-as-entanglement-witness construction (module sync_entanglement_witness) provides the bridge: for separable states, \(R \leq R_{\text{sep}}\), so exceeding the separable bound simultaneously certifies both synchronisation and entanglement.

References¶

Horodecki, M., Horodecki, P. & Horodecki, R. (1996). Phys. Lett. A 223, 1.
Galve, F. et al. (2013). Sci. Rep. 3, 1.
Šotek, M. (2025). God of the Math — The SCPN Master Publications.

Code: scpn_quantum_control.analysis.sync_witness