Average Mixing in Quantum Walks of Reversible Markov Chains

TL;DR

This paper introduces a new formulation of the average mixing matrix for Szegedy quantum walks, linking it to the spectral properties of the underlying Markov chain and comparing it with continuous quantum walks.

Contribution

It defines a new average mixing matrix for Szegedy quantum walks, derives a spectral decomposition formula, and relates its behavior to continuous quantum walks.

Findings

Average uniform mixing in continuous walks implies the same in Szegedy walks.

Provided examples of large Markov chains with uniform mixing in both models.

Abstract

The Szegedy quantum walk is a discrete time quantum walk model which defines a quantum analogue of any Markov chain. The long-term behavior of the quantum walk can be encoded in a matrix called the average mixing matrix, whose columns give the limiting probability distribution of the walk given an initial state. We define a version of the average mixing matrix of the Szegedy quantum walk which allows us to more readily compare the limiting behavior to that of the chain it quantizes. We prove a formula for our mixing matrix in terms of the spectral decomposition of the Markov chain and show a relationship with the mixing matrix of a continuous quantum walk on the chain. In particular, we prove that average uniform mixing in the continuous walk implies average uniform mixing in the Szegedy walk. We conclude by giving examples of Markov chains of arbitrarily large size which admit average uniform mixing in both the continuous and Szegedy quantum walk.