Efficient Quantum Walk Circuits for Metropolis-Hastings Algorithm

TL;DR

This paper develops efficient quantum circuits for the Metropolis-Hastings algorithm, reformulating the quantum walk to avoid costly oracles and demonstrating potential polynomial speedups in heuristic optimization tasks.

Contribution

It introduces a new implementation of Szegedy's quantum walk that bypasses complex oracles and applies heuristic quantum algorithms to optimization problems.

Findings

Numerical results show polynomial quantum speedups in heuristic optimization.

Reformulated quantum walk closely follows classical Metropolis-Hastings walk.

Avoids costly arithmetic operations in quantum oracle implementation.

Abstract

We present a detailed circuit implementation of Szegedy's quantization of the Metropolis-Hastings walk. This quantum walk is usually defined with respect to an oracle. We find that a direct implementation of this oracle requires costly arithmetic operations and thus reformulate the quantum walk in a way that circumvents the implementation of that specific oracle and which closely follows the classical Metropolis-Hastings walk. We also present heuristic quantum algorithms that use the quantum walk in the context of discrete optimization problems and numerically study their performances. Our numerical results indicate polynomial quantum speedups in heuristic settings.