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

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 For random 3-SAT, where such classical methods are no longer applicable, the hybrid MCMC still attains approximately uniform sampling.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.

💡 Research Summary

This paper investigates whether hybrid quantum‑classical Markov‑chain Monte Carlo (MCMC) algorithms can achieve fair sampling of degenerate ground‑state configurations in combinatorial optimization problems. The authors focus on the well‑known bias of quantum optimization heuristics—quantum annealing (QA) and the quantum approximate optimization algorithm (QAOA)—when a simple transverse‑field driver (H_d = ‑∑σ_i^x) is used. Such bias manifests as an exponential suppression of certain ground states due to quantum interference, making uniform sampling impossible with the quantum dynamics alone.

The proposed solution is to treat the quantum dynamics solely as a proposal mechanism within a Metropolis–Hastings framework, while enforcing detailed balance through a classical acceptance step. Two concrete hybrid schemes are examined: (i) Quantum‑enhanced MCMC (Qe‑MCMC), which uses a symmetric unitary U to generate proposals directly from short‑time quantum evolution, and (ii) QAOA‑assisted Neural Monte Carlo (QAOA‑NMC), which trains an autoregressive neural network (MADE) on samples from a depth‑p QAOA circuit and then uses the trained model to produce proposals. In both cases the acceptance probability A(σ′|σ)=min