Enabling Quantum Speedup of Markov Chains using a Multi-level Approach

TL;DR

This paper introduces a multi-level method to accelerate the mixing of Markov chains using quantum computing, reducing complexity from linear in the number of levels to constant.

Contribution

It presents a novel multi-level approach that constructs a sequence of Markov chains with varying resolution, enabling faster quantum mixing.

Findings

Complexity is reduced to O(1) in chain length.

The density of low-resolution chains can warm start high-resolution chains.

The method achieves quantum speedup for Markov chain mixing.

Abstract

Quantum speedup for mixing a Markov chain can be achieved based on the construction of slowly-varying $r$ Markov chains where the initial chain can be easily prepared and the spectral gaps have uniform lower bound. The overall complexity is proportional to $r$. We present a multi-level approach to construct such a sequence of $r$ Markov chains by varying a resolution parameter $h.$ We show that the density function of a low-resolution Markov chain can be used to warm start the Markov chain with high resolution. We prove that in terms of the chain length the new algorithm has $O(1)$ complexity rather than $O(r).$