Fair sampling of ground-state configurations using hybrid quantum-classical MCMC algorithms

TL;DR

This paper demonstrates that hybrid quantum-classical MCMC algorithms can effectively correct quantum sampling biases, achieving near-uniform sampling over degenerate ground states in combinatorial optimization problems.

Contribution

The study introduces a hybrid quantum-classical MCMC framework that restores fair sampling in quantum heuristics, validated on Ising models and random SAT problems.

Findings

MCMC post-processing corrects quantum sampling bias.

Hybrid MCMC achieves near-uniform sampling in Ising models.

Effective solution counting with comparable transition numbers to classical methods.

Abstract

We study the fair sampling properties of hybrid quantum-classical Markov chain Monte Carlo (MCMC) algorithms for combinatorial optimization problems with degenerate ground states. While quantum optimization heuristics such as quantum annealing and the quantum approximate optimization algorithm (QAOA) are known to induce biased sampling, hybrid quantum-classical MCMC incorporates quantum dynamics only as a proposal transition and enforces detailed balance through classical acceptance steps. Using small Ising models, we show that MCMC post-processing corrects the sampling bias of quantum dynamics and restores near-uniform sampling over degenerate ground states. We then apply the method to random $k$-SAT problems near the satisfiability threshold. For random 2-SAT, a hybrid MCMC combining QAOA-assisted neural proposals with single spin-flip updates achieves fairness comparable to that of PT-ICM. For random 3-SAT, where such classical methods are no longer applicable, the hybrid MCMC still attains approximately uniform sampling. We also examine solution counting and find that the required number of transitions is comparable to that of WalkSAT. These results indicate that hybrid quantum-classical MCMC provides a viable framework for fair sampling and solution enumeration.