Average mixing of continuous quantum walks

TL;DR

This paper studies the average mixing matrix derived from continuous quantum walks on graphs, revealing its properties and simple forms for paths and cycles, with implications for quantum computing.

Contribution

It establishes fundamental properties of the average mixing matrix, including positivity, rational entries, and explicit forms for specific graph classes.

Findings

The average mixing matrix is positive semidefinite.

Entries of the matrix are always rational.

For paths and cycles, the matrix has a simple, explicit form.

Abstract

If $X$ is a graph with adjacency matrix $A$, then we define $H(t)$ to be the operator $\exp(itA)$. The Schur (or entrywise) product $H(t)\circ H(-t)$ is a doubly stochastic matrix and, because of work related to quantum computing, we are concerned the \textsl{average mixing matrix}. This can be defined as the limit of $C^{-1} \int_0^C H(t)\circ H(-t)\dt$ as $C\to\infty$. We establish some of the basic properties of this matrix, showing that it is positive semidefinite and that its entries are always rational. We find that for paths and cycles this matrix takes on a surprisingly simple form, thus for the path it is a linear combination of $I$, $J$ (the all-ones matrix), and a permutation matrix.