Abstract Markov chain Monte Carlo (MCMC) methods are a family of algorithms designed to approximate probability distributions that are otherwise intractable to sample from directly. However, as the dimensionality and complexity of the target distribution increase, the convergence of these methods can slow considerably, demanding substantial computational resources. In this context, quantum computing emerges as a promising framework to address this limitation, with different quantum circuit constructions serving as quantum MCMC methods. In this work, we present a quantum circuit based on the discrete quantum walk (DQW) algorithm that implements the algorithmic steps of the Metropolis–Hastings method, encoding the target distribution in a position quantum register. The circuit incorporates auxiliary registers to compute and discretize the acceptance probability, and uses controlled operations to modulate the walker’s evolution accordingly. Simulation results show that the marginal probability distribution encoded in the position register converges asymptotically toward the discretized target distribution for both single and multimodal distributions. We further analyze the effect of circuit parameters on convergence behavior and approximation quality, refining the construction by enabling the evaluation of multiple candidate moves per iteration, therefore reducing the number of iterations required for convergence within our implementation. Finally, we provide a qualitative comparison of our approach with related quantum MCMC methods, namely Szegedy’s quantum walk and QAOA, and discuss structural differences as well as the potential incorporation of adaptive proposal techniques within the proposed circuit framework.
Abstract Szegedy’s quantization of a reversible Markov chain provides a quantum walk whose spectral gap is quadratically larger than that of the classical walk. Quantum computers are therefore expected to provide a speedup of Metropolis–Hastings (MH) simulations. Existing generic methods to implement the quantum walk require coherently computing the transition probabilities of the underlying Markov kernel. However, reversible computing methods require a number of qubits that scales with the complexity of the computation. This overhead is undesirable in near-term fault-tolerant quantum computing, where few logical qubits are available. In this work, we present a Szegedy quantum walk construction which follows the classical proposal-acceptance logic, and does not require further reversible computing methods. We also compare this construction with an alternative to Szegedy’s approach which also provides a quadratic gap amplification. Since each step of the quantum walks uses a constant number of proposal and acceptance steps, we expect the end-to-end quadratic speedup to hold for MH Markov Chain Monte-Carlo simulations.
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
The Metropolis-Hastings algorithm is a cornerstone of Markov Chain Monte Carlo methods, underpinning a wide range of applications in computational physics, Bayesian inference, and machine learning. Quantum variants of Metropolis-Hastings promise accelerated mixing through quantum walks, but their practical realisation remains challenging. In this work, we construct and simulate an explicit circuit level implementation of a quantum Metropolis-Hastings algorithm based on the framework introduced by Claudon \emph{et al.} (arXiv:2506.11576). We present the full quantum workflow required to prepare a stationary distribution, including a number of modifications required to make the algorithm implementable in a realistic quantum circuit model. Our results demonstrate that these modifications are essential to recover the correct stationary behaviour and highlight both the potential and current limitations of quantum Metropolis-Hastings algorithms, which are expected to become practically relevant in the fault tolerant quantum computing regime.
The Markov chain Monte Carlo (MCMC) method, especially the Metropolis-Hastings (MH) algorithm, is a widely used technique for sampling from a target probability distribution
P on a state space $\mathrm{\ensuremath{\Omega}}$ and applied to various problems such as estimation of parameters in statistical models in the Bayesian approach. Quantum algorithms for MCMC have been proposed, yielding the quadratic speedup with respect to the spectral gap $\mathrm{\ensuremath{\Delta}}$ compared to classical counterparts. In this paper, we consider the quantum version of the MH algorithm in the case that calculating
P is costly because the log-likelihood
L for a state $x\ensuremath{\in}\mathrm{\ensuremath{\Omega}}$ is obtained via computing the sum of many terms, $\frac{1}{M}{\ensuremath{\sum}}_{i=0}^{M\ensuremath{-}1}\ensuremath{\ell}(i,x)$. We propose calculating
L by quantum Monte Carlo integration and combine it with the existing method called quantum simulated annealing (QSA) to generate the quantum state that encodes
P in amplitudes. We consider not only state generation but also finding a credible interval for a parameter, a common task in Bayesian inference. In the proposed method for credible interval calculation, the number of queries to the quantum circuit to compute $\ensuremath{\ell}$ scales on $\mathrm{\ensuremath{\Delta}}$, the required accuracy $\ensuremath{\epsilon}$, and the standard deviation $\ensuremath{\sigma}$ of $\ensuremath{\ell}$ as $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{O}(\ensuremath{\sigma}/{\ensuremath{\epsilon}}^{2}{\mathrm{\ensuremath{\Delta}}}^{3/2})$, in contrast to $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{O}(M/\ensuremath{\epsilon}{\mathrm{\ensuremath{\Delta}}}^{1/2})$ for QSA with
L calculated exactly. Therefore, the proposed method is advantageous if $\ensuremath{\sigma}$ scales on
M sublinearly. As one such example, we consider parameter estimation in a gravitational wave experiment, where $\ensuremath{\sigma}=O({M}^{1/2})$.
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.
We present an implementation of Quantum Computing for a Markov Chain Monte Carlo method with an application to cosmological functions, to derive posterior distributions from cosmological probes. The algorithm proposes new steps in the parameter space via a quantum circuit whose resulting statevector provides the components of the shift vector. The proposed point is accepted or rejected via the classical Metropolis-Hastings acceptance method. The advantage of this hybrid quantum approach is that the step size and direction change in a way independent of the evolution of the chain, thus ideally avoiding the presence of local minima. The results are consistent with analyses performed with classical methods, both for a test function and real cosmological data. The final goal is to generalize this algorithm to test its application to complex cosmological computations.