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arXiv:2604.20701

Divide-and-Conquer Neural Network Surrogates for Quantum Sampling: Accelerating Markov Chain Monte Carlo in Large-Scale Constrained Optimization Problems

Yuya Kawamata, Yuichiro Nakano, Keisuke Fujii

quant-phSubmitted Apr 22, 2026• Announced Apr 23, 2026• 0 citations
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Abstract

Sampling problems are promising candidates for demonstrating quantum advantage, and one approach known as quantum-enhanced Markov chain Monte Carlo [Layden, D. et al., Nature 619, 282-287 (2023)] uses quantum samples as a proposal distribution to accelerate convergence to a target distribution. On the other hand, many practical problems are large-scale and constrained, making it difficult to construct efficient proposal distributions in classical methods and slowing down MCMC mixing. In this work, we propose a divide-and-conquer neural network surrogate framework for quantum sampling to accelerate MCMC under fixed Hamming weight constraints. Our method divides the interaction graph for an Ising problem into subgraphs, generates samples using QAOA for those subproblems with an XY mixer, and trains neural network surrogates conditioned on the Hamming weight to provide proposal distributions for each subset while preserving the constraint. In numerical experiments of Boltzmann sampling on 3-regular graphs, our method consistently accelerated mixing as the system size $N$ increased, with average improvements in the autocorrelation decay rate constant by speedup factors of about $20.3$ and $7.6$ over classical pair-flip methods based on nearest-neighbor and non-nearest-neighbor exchanges, respectively. We also applied the method to an MNIST feature mask optimization problem with $N=784$, obtaining faster energy convergence and a $2.03\%$ higher classification accuracy. These results show that our method enables efficient and scalable MCMC and can outperform classical methods for practical applications on NISQ devices.

Comments
12 pages, 9 figures

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Sampling problems are promising candidates for demonstrating quantum advantage, and one approach known as quantum-enhanced Markov chain Monte Carlo [Layden, D. et al., Nature 619, 282-287 (2023)] uses quantum samples as a proposal distribution to accelerate convergence to a target distribution. On the other hand, many practical problems are large-scale and constrained, making it difficult to construct efficient proposal distributions in classical methods and slowing down MCMC mixing. In this work, we propose a divide-and-conquer neural network surrogate framework for quantum sampling to accelerate MCMC under fixed Hamming weight constraints. Our method divides the interaction graph for an Ising problem into subgraphs, generates samples using QAOA for those subproblems with an XY mixer, and trains neural network surrogates conditioned on the Hamming weight to provide proposal distributions for each subset while preserving the constraint. In numerical experiments of Boltzmann sampling on 3-regular graphs, our method consistently accelerated mixing as the system size NNN increased, with average improvements in the autocorrelation decay rate constant by speedup factors of about 20.320.320.3 and 7.67.67.6 over classical pair-flip methods based on nearest-neighbor and non-nearest-neighbor exchanges, respectively. We also applied the method to an MNIST feature mask optimization problem with N=784N=784N=784, obtaining faster energy convergence and a 2.03%2.03\%2.03% higher classification accuracy. These results show that our method enables efficient and scalable MCMC and can outperform classical methods for practical applications on NISQ devices.
Sampling problems are widely regarded as the task for which quantum computers can most readily provide a quantum advantage. Leveraging this feature, the quantum-enhanced Markov chain Monte Carlo [Layden , Nature 619, 282–287 (2023)] has been proposed recently, where sampling from a quantum computer is used as a proposal distribution and convergence to a target distribution is accelerated. However, guaranteeing convergence to the target distribution typically forces one to impose restrictive symmetry constraints on the quantum circuit, which makes it hard to design good proposal distributions and prevents making full use of the advantage of a quantum computer. We explore a hybrid quantum-classical Markov chain Monte Carlo framework that combines a quantum circuit with a generative neural sampler (GNS). The GNS is trained on quantum samples and acts as a classical surrogate to efficiently emulate quantum outputs, thereby lifting circuit constraints. We apply this method to Boltzmann sampling of spin glasses using proposals trained with a quantum approximate optimization algorithm circuit. This approach outperforms conventional methods, showing a polynomial speedup (over cubic) in spectral gap compared with uniform proposals. Notably, it maintains similar acceleration even without parameter optimization. These results establish the method as a viable sampling-based quantum algorithm for noisy intermediate-scale quantum devices and highlight its potential for solving practical problems with quantum computation.
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Quantum computation is expected to accelerate certain computational tasks over classical counterparts. Its most primitive advantage is its ability to sample from classically intractable probability distributions. A promising approach to make use of this fact is the so-called quantum-enhanced Markov-chain Monte Carlo (qe-MCMC) method [D. Layden , ], which uses outputs from quantum circuits as the proposal distributions. In this paper, we propose the use of a quantum alternating operator ansatz (QAOA) for qe-MCMC and provide a strategy to optimize its parameters to improve convergence speed while keeping its depth shallow. The proposed QAOA-type circuit is designed to satisfy the specific constraint which qe-MCMC requires with arbitrary parameters. Through our extensive numerical analysis, we find a correlation in a certain parameter range between an experimentally measurable value, acceptance rate of MCMC, and the spectral gap of the MCMC transition matrix, which determines the convergence speed. This allows us to optimize the parameter in the QAOA circuit and achieve quadratic speedup in convergence. Since MCMC is used in various areas such as statistical physics and machine learning, this paper represents an important step toward realizing practical quantum advantage with currently available quantum computers through qe-MCMC. Published by the American Physical Society 2024
Sampling from complicated probability distributions is a hard computational problem arising in many fields, including statistical physics, optimization, and machine learning. Quantum computers have recently been used to sample from complicated distributions that are hard to sample from classically, but which seldom arise in applications. Here we introduce a quantum algorithm to sample from distributions that pose a bottleneck in several applications, which we implement on a superconducting quantum processor. The algorithm performs Markov chain Monte Carlo (MCMC), a popular iterative sampling technique, to sample from the Boltzmann distribution of classical Ising models. In each step, the quantum processor explores the model in superposition to propose a random move, which is then accepted or rejected by a classical computer and returned to the quantum processor, ensuring convergence to the desired Boltzmann distribution. We find that this quantum algorithm converges in fewer iterations than common classical MCMC alternatives on relevant problem instances, both in simulations and experiments. It therefore opens a new path for quantum computers to solve useful--not merely difficult--problems in the near term.
Sampling tasks are a natural class of problems for quantum computers due to the probabilistic nature of the Born rule. Sampling from useful distributions on noisy quantum hardware remains a challenging problem. A recent paper [Layden, D. et al. Nature 619, 282-287 (2023)] proposed a quantum-enhanced Markov chain Monte Carlo algorithm where moves are generated by a quantum device and accepted or rejected by a classical algorithm. While this procedure is robust to noise and control imperfections, its potential for quantum advantage is unclear. Here we show that there is no speedup over classical sampling on a worst-case unstructured sampling problem. We present an upper bound to the Markov gap that rules out a speedup for any unital quantum proposal.
Quantum computing offers an alternative paradigm for addressing combinatorial optimization problems compared to classical computing. Despite recent hardware improvements, the execution of empirical quantum optimization experiments at scales known to be hard for state-of-the-art classical solvers is not yet in reach. In this work, we offer a different way to approach combinatorial optimization with near-term quantum computing. Motivated by the promising results observed in using quantum-enhanced Markov chain Monte Carlo (QeMCMC) for approximating complicated probability distributions, we combine ideas of sampling from the device with QeMCMC together with warm-starting and parallel tempering, in the context of combinatorial optimization. We demonstrate empirically that our algorithm recovers the global optima for instances of the Maximum Independent Set problem (MIS) up to 117 decision variables using 117 qubits on IBM quantum hardware. We show early evidence of a scaling advantage of our algorithm compared to similar classical methods for the chosen instances of MIS. MIS is practically relevant across domains like financial services and molecular biology, and, in some cases, already difficult to solve to optimality classically with only a few hundred decision variables.