Space-efficient Quantization Method for Reversible Markov Chains

TL;DR

This paper introduces a space-efficient quantum walk construction for certain reversible Markov chains, reducing ancilla requirements by leveraging symmetric proposals and energy-based encoding.

Contribution

It presents a novel quantization method that minimizes ancilla space by encoding energy levels instead of the entire state space, improving quantum sampling efficiency.

Findings

Reduces ancilla space from state space size to number of energy levels.

Develops a block encoding technique for Hadamard products of matrices.

Applicable to Markov chains with symmetric proposals and Gibbs distribution sampling.

Abstract

In a seminal paper, Szegedy showed how to construct a quantum walk $W(P)$ for any reversible Markov chain $P$ such that its eigenvector with eigenphase $0$ is a quantum sample of the limiting distribution of the random walk and its eigenphase gap is quadratically larger than the spectral gap of $P$. The standard construction of Szegedy's quantum walk requires an ancilla register of Hilbert-space dimension equal to the size of the state space of the Markov chain. We show that it is possible to avoid this doubling of state space for certain Markov chains that employ a symmetric proposal probability and a subsequent accept/reject probability to sample from the Gibbs distribution. For such Markov chains, we give a quantization method which requires an ancilla register of dimension equal to only the number of different energy values, which is often significantly smaller than the size of the state space. To accomplish this, we develop a technique for block encoding Hadamard products of matrices which may be of wider interest.