A quantum parallel Markov chain Monte Carlo

TL;DR

This paper introduces a hybrid quantum-classical approach to parallel MCMC that leverages quantum algorithms to significantly reduce the number of target density evaluations needed, enhancing efficiency in sampling methods.

Contribution

It presents a novel hybrid quantum strategy for parallel MCMC that reduces target evaluations from linear to square root order using quantum search techniques.

Findings

Reduces target evaluations from O(P) to O(√P)

Integrates Gumbel-max trick with quantum search algorithms

Provides a framework for quantum-accelerated MCMC sampling

Abstract

We propose a novel hybrid quantum computing strategy for parallel MCMC algorithms that generate multiple proposals at each step. This strategy makes the rate-limiting step within parallel MCMC amenable to quantum parallelization by using the Gumbel-max trick to turn the generalized accept-reject step into a discrete optimization problem. When combined with new insights from the parallel MCMC literature, such an approach allows us to embed target density evaluations within a well-known extension of Grover's quantum search algorithm. Letting $P$ denote the number of proposals in a single MCMC iteration, the combined strategy reduces the number of target evaluations required from $\mathcal{O}(P)$ to $\mathcal{O}(P^{1/2})$. In the following, we review the rudiments of quantum computing, quantum search and the Gumbel-max trick in order to elucidate their combination for as wide a readership as possible.