Unexpected Averages of Mixing Matrices

TL;DR

This paper explores how different probability distributions affect the average mixing matrix in continuous-time quantum walks, revealing unexpected behaviors and potential for controlling quantum effects.

Contribution

It extends the analysis of average mixing matrices to general distributions and uncovers surprising cases where matrices have constant entries, suggesting new ways to influence quantum walks.

Findings

Algebraic properties of average mixing matrices hold for general distributions.

Existence of distributions leading to constant-entry average mixing matrices.

Connections between trace of the average mixing matrix and quantum walk properties.

Abstract

The (standard) average mixing matrix of a continuous-time quantum walk is computed by taking the expected value of the mixing matrices of the walk under the uniform sampling distribution on the real line. In this paper we consider alternative probability distributions, either discrete or continuous, and first we show that several algebraic properties that hold for the average mixing matrix still stand for this more general setting. Then, we provide examples of graphs and choices of distributions where the average mixing matrix behaves in an unexpected way: for instance, we show that there are probability distributions for which the average mixing matrices of the paths on three or four vertices have constant entries, opening a significant line of investigation about how to use classical probability distributions to sample quantum walks and obtain desired quantum effects. We present results connecting the trace of the average mixing matrix and quantum walk properties, and we show that the Gram matrix of average states is the average mixing matrix of a certain related distribution. Throughout the text, we employ concepts of classical probability theory not usually seen in texts about quantum walks.