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arXiv:2610.36431

Fixed-Horizon Error Rates for Array-RQMC Metropolis-Hastings

Jiarui Du, Josef Dick, Zexin Pan, Houying Zhu

Published Oct 27, 2026• 0 citations
arXivHTMLPDF

Abstract

This paper studies randomised quasi-Monte Carlo (RQMC) methods for Metropolis--Hastings (MH) algorithms, whose accept--reject discontinuity complicates RQMC analysis. To circumvent some of these complications, we study Array-RQMC for MH, with $N$ replicated chains evolved over a fixed time horizon $T$, where we make use of Hilbert sorting for joint multidimensional updates at each step. For one-dimensional MH and bounded-variation test functions, we prove a sampling mean squared error bound of $\mathcal O_T(N^{-4/3})$ relative to the finite-horizon expectation. For product targets, coordinatewise Array-RQMC Metropolis-within-Gibbs retains this exponent for additive marginal bounded-variation tests, with constants that may depend on dimension. With Gaussian random-walk proposals, the bounds extend to scalar and additive marginal polynomials with an additional factor $(\log N)^\ell$, where $\ell$ is the maximum polynomial degree. For joint updates in state dimension $p$ using $p$ proposal uniforms, we provide a conditional $\mathcal O_T(N^{-1-1/(p+2)})$ framework under regularity assumptions that are not verified for the joint multidimensional examples. Numerical experiments support the verified one-dimensional theory and demonstrate variance reduction for joint Hilbert-sorted updates, including Bayesian logistic regression, while illustrating the limitations of these gains as dimension increases.

DOI
10.48550/arxiv.2610.36431

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Randomized quasi-Monte Carlo (RQMC) methods estimate the expectation of a random variable by the average of n dependent realizations of it. In general, due to the strong dependence, the estimation error may not obey a central limit theorem. Analysis of RQMC methods have so far focused mostly on the convergence rates of asymptotic worst-case error bounds and variance bounds, when n → ∞, but little is known about the limiting distribution of the error. Here we examine this limiting distribution for the special case of a randomly-shifted lattice rule, when the integrand is smooth. We start with simple one-dimensional functions, where we show that the limiting distribution is uniform over a bounded interval if the integrand is non-periodic, and has a square root form over a bounded interval if the integrand is periodic. In higher dimensions, for linear functions, the distribution function of the properly standardized error converges to a spline of degree equal to the dimension.
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 $\mathrm{\ensuremath{\Omega}}$ 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 PPP is costly because the log-likelihood LLL 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 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 $\ensuremath{\ell}$ 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 LLL calculated exactly. Therefore, the proposed method is advantageous if $\ensuremath{\sigma}$ scales on MMM sublinearly. As one such example, we consider parameter estimation in a gravitational wave experiment, where $\ensuremath{\sigma}=O({M}^{1/2})$.