Non-uniform mixing of quantum walk on cycles

TL;DR

This paper demonstrates that continuous-time quantum walks on cycles and other Abelian circulant graphs do not converge to a uniform distribution, contrasting classical random walks which do mix uniformly.

Contribution

It reveals the non-uniform mixing behavior of quantum walks on cycles and Abelian circulant graphs, a significant departure from classical mixing properties.

Findings

Quantum walks on most even cycles are never uniformly mixed at any instant.

Average distribution of quantum walks on Abelian circulant graphs is never uniform.

Results apply to standard circulant graphs like cycles, complete graphs, and hypercubes.

Abstract

A classical lazy random walk on cycles is known to mix to the uniform distribution. In contrast, we show that a continuous-time quantum walk on cycles exhibit strong non-uniform mixing properties. Our results include the following: - The instantaneous distribution of a quantum walk on most even-length cycles is never uniform. - The average distribution of a quantum walk on any Abelian circulant graph is never uniform. As a corollary, the average distribution of a quantum walk on any standard circulant graph, such as the cycles, complete graphs, and even hypercubes, is never uniform.