Quantum Circuits for the Metropolis-Hastings Algorithm

TL;DR

This paper introduces a new quantum walk construction for the Metropolis-Hastings algorithm that avoids complex reversible computing, potentially enabling quadratic speedups in quantum MCMC simulations.

Contribution

It presents a Szegedy quantum walk method that follows classical acceptance logic without requiring reversible computing, simplifying implementation for near-term quantum computers.

Findings

Proposes a quantum walk construction aligned with classical MH logic.

Achieves quadratic spectral gap amplification without reversible computing overhead.

Expected to enable quadratic speedup in quantum MCMC simulations.

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.