arXiv:2606.23727 · v3
Pauli-Z Moment Matrices in the Variational Quantum Power Method: Goemans-Williamson Rounding, Qubit Locking, and a Correlation-Based Circuit-Cutting Heuristic
摘要
The matrix of single- and two-qubit Pauli-Z expectation values of any quantum state is the moment matrix of its computational-basis outcome distribution; it therefore lies in the cut polytope and, in particular, is a feasible point of the degree-2 Sum-of-Squares (Goemans-Williamson, GW) relaxation. Applying GW random-hyperplane rounding to this matrix yields cuts of expected weight at least alpha_GW times the state's energy. This observation and the resulting bound are not new: the bound is the GW inequality applied to the circuit's covariance matrix, and it was stated and used in exactly this form by Martinez, Fawzi and Stilck Franca, who interpret it as showing that a classical surrogate nearly reproduces the circuit and add noisy-circuit guarantees and hardware experiments; relax-and-round of QAOA correlation matrices, with spectral rather than hyperplane rounding, was introduced earlier by Dupont and Sundar. The bound is also not stronger than what direct computational-basis sampling of the state already gives in expectation. In this note we revisit the bound as a post-processing of the two-point data produced by variational algorithms such as QAOA and the Variational Quantum Power Method (VQPM), and give an energy-leakage bound for VQPM qubit locking. We further discuss a heuristic that uses the connected Z-correlation matrix to propose candidate qubit bipartitions for circuit cutting, in the spirit of existing correlation- and mutual-information-based partitioning methods, together with product-state error bounds in terms of quantum mutual information. Pairwise Z-correlations neither certify product structure of the state nor factorization of the underlying unitary, and the error bounds require quantities not available from Z-data alone.
作者备注：v3: retitled and substantially revised to credit prior work. Thms 2.1, 2.3 are stated in Martinez, Fawzi, Stilck Franca (arXiv:2507.21883; Quantum 10, 2068 (2026)); relax-and-round of QAOA correlation matrices with spectral rounding is due to Dupont & Sundar (arXiv:2307.05821; PRA 109, 012429 (2024))