Quantum Walks on Necklaces and Mixing

TL;DR

This paper studies continuous-time quantum walks on necklace graphs, demonstrating how their mixing times scale similarly to classical walks on cycles, using a Bloch ansatz to simplify analysis.

Contribution

It introduces a general method for analyzing quantum walk mixing times on necklace graphs, reducing the problem to the size of a single pearl and comparing quantum and classical mixing behaviors.

Findings

Mixing times scale similarly to classical cycles

Bloch ansatz simplifies eigenfunction analysis

Results apply to various necklace graph configurations

Abstract

We analyze continuous-time quantum walks on necklace graphs - cyclical graphs consisting of many copies of a smaller graph (pearl). Using a Bloch-type ansatz for the eigenfunctions, we block-diagonalize the Hamiltonian, reducing the effective size of the problem to the size of a single pearl. We then present a general approach for showing that the mixing time scales (with growing size of the necklace) similarly to that of a simple walk on a cycle. Finally, we present results for mixing on several necklace graphs.