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Average Mixing in Quantum Walks of Reversible Markov Chains

Julien Sorci

TL;DR

This work studies the long-term behavior of the Szegedy discrete-time quantum walk associated with a Markov chain by introducing and analyzing the average mixing matrix $\widehat{M}$. The authors derive a closed-form expression for $\widehat{M}$ in terms of the spectral idempotents of the chain's discriminant $D$, and relate it to the average mixing matrix of the corresponding continuous quantum walk $\widehat{M}_C$, establishing $\widehat{M} = \widehat{M}_C - \tfrac{1}{2}(I - P^T)\sum_{r=2}^m \frac{1}{1-\lambda_r^2} E_r^{\circ 2}$. They prove that average uniform mixing in the continuous walk implies average uniform mixing in the Szegedy walk, and provide an infinite family of symmetric Markov chains of arbitrary size that exhibit average uniform mixing in both models via a tensor-product construction with carefully chosen eigenvalues. The results advance quantum sampling theory by giving concrete spectral criteria and constructive examples for when quantum walks yield uniform limiting distributions over vertices, with potential implications for speedups in sampling and counting tasks on weighted graphs.

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.

Average Mixing in Quantum Walks of Reversible Markov Chains

TL;DR

This work studies the long-term behavior of the Szegedy discrete-time quantum walk associated with a Markov chain by introducing and analyzing the average mixing matrix . The authors derive a closed-form expression for in terms of the spectral idempotents of the chain's discriminant , and relate it to the average mixing matrix of the corresponding continuous quantum walk , establishing . They prove that average uniform mixing in the continuous walk implies average uniform mixing in the Szegedy walk, and provide an infinite family of symmetric Markov chains of arbitrary size that exhibit average uniform mixing in both models via a tensor-product construction with carefully chosen eigenvalues. The results advance quantum sampling theory by giving concrete spectral criteria and constructive examples for when quantum walks yield uniform limiting distributions over vertices, with potential implications for speedups in sampling and counting tasks on weighted graphs.

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.
Paper Structure (7 sections, 10 theorems, 33 equations)

This paper contains 7 sections, 10 theorems, 33 equations.

Table of Contents

  1. Introduction
  2. The Szegedy Quantum Walk
  3. The Average Mixing Matrix
  4. Continuous Quantum Walks
  5. Properties of $\widehat{M}$
  6. Examples of Average Uniform Mixing
  7. Discussion and Open Questions

Key Result

Lemma 1

Let $U$ be the transition matrix for the Szegedy quantum walk of an arbitrary Markov chain $P$ with discriminant $D$. For each eigenvalue $\lambda$ of $D$ let $E_\lambda$ denote the projection onto the $\lambda$-eigenspace of $D$, and write $\lambda = \cos\theta_\lambda$. Then

Theorems & Definitions (18)

  • Lemma 1
  • Lemma 2
  • proof
  • Theorem 3
  • proof
  • Lemma 4
  • Theorem 5
  • proof
  • Theorem 6
  • proof
  • ...and 8 more