Quantum Metropolis-Hastings via Penalised Qubitized Walks: Spectral Filtering and Circuit Implementation

TL;DR

This paper develops a circuit-level implementation of quantum Metropolis-Hastings, addressing practical challenges and demonstrating its potential and limitations for future fault-tolerant quantum computing.

Contribution

It provides a detailed quantum circuit framework for Metropolis-Hastings, including modifications for realistic implementation and analysis of its stationary distribution.

Findings

Modifications are essential for correct stationary behaviour.

The implementation highlights potential and limitations in fault-tolerant regimes.

Simulation results demonstrate the algorithm's practical feasibility.

Abstract

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.