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arXiv:2303.05640 · Physical Review Research

Quantum Metropolis-Hastings algorithm with the target distribution calculated by quantum Monte Carlo integration

Koichi Miyamoto

Published Jul 28, 2023• 2 citations
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Abstract

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 $\ensuremath{\sigma}=O({M}^{1/2})$.

DOI
10.1103/physrevresearch.5.033059

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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 PPP on a state space \ensuremathΩ\mathrm{\ensuremath{\Omega}}\ensuremathΩ 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 \ensuremathΔ\mathrm{\ensuremath{\Delta}}\ensuremathΔ compared to classical counterparts. In this paper, we consider the quantum version of the MH algorithm in the case that calculating PPP is costly because the log-likelihood LLL for a state x\ensuremath∈\ensuremathΩx\ensuremath{\in}\mathrm{\ensuremath{\Omega}}x\ensuremath∈\ensuremathΩ is obtained via computing the sum of many terms, 1M\ensuremath∑i=0M\ensuremath−1\ensuremathℓ(i,x)\frac{1}{M}{\ensuremath{\sum}}_{i=0}^{M\ensuremath{-}1}\ensuremath{\ell}(i,x)M1​\ensuremath∑i=0M\ensuremath−1​\ensuremathℓ(i,x). We propose calculating LLL by quantum Monte Carlo integration and combine it with the existing method called quantum simulated annealing (QSA) to generate the quantum state that encodes PPP 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 \ensuremathℓ\ensuremath{\ell}\ensuremathℓ scales on \ensuremathΔ\mathrm{\ensuremath{\Delta}}\ensuremathΔ, the required accuracy \ensuremathϵ\ensuremath{\epsilon}\ensuremathϵ, and the standard deviation \ensuremathσ\ensuremath{\sigma}\ensuremathσ of \ensuremathℓ\ensuremath{\ell}\ensuremathℓ as O\ifmmode~\else˜\fi(\ensuremathσ/\ensuremathϵ2\ensuremathΔ3/2)\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{O}(\ensuremath{\sigma}/{\ensuremath{\epsilon}}^{2}{\mathrm{\ensuremath{\Delta}}}^{3/2})O\ifmmode~\else˜\fi​(\ensuremathσ/\ensuremathϵ2\ensuremathΔ3/2), in contrast to O\ifmmode~\else˜\fi(M/\ensuremathϵ\ensuremathΔ1/2)\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{O}(M/\ensuremath{\epsilon}{\mathrm{\ensuremath{\Delta}}}^{1/2})O\ifmmode~\else˜\fi​(M/\ensuremathϵ\ensuremathΔ1/2) for QSA with LLL calculated exactly. Therefore, the proposed method is advantageous if \ensuremathσ\ensuremath{\sigma}\ensuremathσ scales on MMM sublinearly. As one such example, we consider parameter estimation in a gravitational wave experiment, where \ensuremathσ=O(M1/2)\ensuremath{\sigma}=O({M}^{1/2})\ensuremathσ=O(M1/2).
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.
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.
The approximation of an integral with numerical integration methods is a relevant problem in many fields, from finance to chemistry. Monte Carlo methods are a family of methods widely used for numerical integration but which are not always applicable to industrial tasks. Instead, there are quantum computing methods such as Quantum Amplitude Estimation able to offer a speed-up compared to the Monte Carlo methods for numerical integration which promise a significant impact, especially in finance. The Quantum Amplitude Estimation algorithm is a fundamental quantum algorithm with the potential to achieve a quadratic speed-up. Unfortunately, being competitive over classical Monte Carlo methods would require hardware performances currently not available. Therefore, alternative methods have been developed in order to exploit fewer resources while maintaining an advantageous theoretical scaling. We present a comparison between the standard QAE algorithm and two NISQ-friendly versions of the QAE on a numerical integration task. The Monte Carlo technique of Metropolis-Hastings instead is used as a classical benchmark. The scaling of the different algorithms are evaluated in terms of estimation error as a function of the number of samples, computational time, and length of the quantum circuits required by the quantum solutions. The conclusion highlights that an alternative approach is preferable with respect to employng the phase estimation routine. Indeed, the maximum likelihood estimation guarantees the best trade-off between the length of the quantum circuits and the precision in the integral estimation.
Copyright © 2020 by SIAM Markov chain Monte Carlo algorithms have important applications in counting problems and in machine learning problems, settings that involve estimating quantities that are difficult to compute exactly. How much can quantum computers speed up classical Markov chain algorithms? In this work we consider the problem of speeding up simulated annealing algorithms, where the stationary distributions of the Markov chains are Gibbs distributions at temperatures specified according to an annealing schedule. We construct a quantum algorithm that both adaptively constructs an annealing schedule and quantum samples at each temperature. Our adaptive annealing schedule roughly matches the length of the best classical adaptive annealing schedules and improves on nonadaptive temperature schedules by roughly a quadratic factor. Our dependence on the Markov chain gap matches other quantum algorithms and is quadratically better than what classical Markov chains achieve. Our algorithm is the first to combine both of these quadratic improvements. Like other quantum walk algorithms, it also improves on classical algorithms by producing “qsamples” instead of classical samples. This means preparing quantum states whose amplitudes are the square roots of the target probability distribution. In constructing the annealing schedule we make use of amplitude estimation, and we introduce a method for making amplitude estimation nondestructive at almost no additional cost, a result that may have independent interest. Finally we demonstrate how this quantum simulated annealing algorithm can be applied to the problems of estimating partition functions and Bayesian inference.
In this study, we propose quantum annealing-enhanced Markov Chain Monte Carlo (QAEMCMC), where QA is integrated into the MCMC subroutine. QA efficiently explores low-energy configurations and overcomes local minima, enabling the generation of proposal states with a high acceptance probability. We benchmark QAEMCMC for the Sherrington-Kirkpatrick model and demonstrate its superior performance over the classical MCMC method. Our results reveal larger spectral gaps, faster convergence of energy observables, and reduced total variation distance between the empirical and target distributions. QAEMCMC accelerates MCMC and provides an efficient method for complex systems, paving the way for scalable quantum-assisted sampling strategies.