Average mixing of continuous quantum walks

8Citations
Citations of this article
13Readers
Mendeley users who have this article in their library.

Abstract

If X is a graph with adjacency matrix A, then we define H(t) to be the operator exp(i t A). The Schur (or entrywise) product H(t) {ring operator} H( - t) is a doubly stochastic matrix and because of work related to quantum computing, we are concerned with the average mixing matrix M̂X, defined by. M̂X=limT→∞1T∫0TH(t){ring operator}H(-t)dt. In this paper we establish some of the basic properties of this matrix, showing that it is positive semidefinite and that its entries are always rational. We see that in a number of cases its form is surprisingly simple. Thus for the path on n vertices it is equal to. 12n+2(2J+I+T) where T is the permutation matrix that swaps j and n + 1 - j for each j. If X is an odd cycle or, more generally, if X is one of the graphs in a pseudocyclic association scheme on n vertices with d classes, each of valency m, then its average mixing matrix is. n-m+1n2J+m-1nI. (One reason this is interesting is that a graph in a pseudocyclic scheme may have trivial automorphism group, and then the mixing matrix is more symmetric than the graph itself.). © 2013 Elsevier Inc.

Cite

CITATION STYLE

APA

Godsil, C. (2013). Average mixing of continuous quantum walks. Journal of Combinatorial Theory. Series A, 120(7), 1649–1662. https://doi.org/10.1016/j.jcta.2013.05.006

Register to see more suggestions

Mendeley helps you to discover research relevant for your work.

Already have an account?

Save time finding and organizing research with Mendeley

Sign up for free