Average mixing of continuous quantum walks

If X is a graph with adjacency matrix A  , then we define H(t)H(t) to be the operator exp(itA)exp(itA). The Schur (or entrywise) product H(t)∘H(−t)H(t)∘H(−t) is a doubly stochastic matrix and because of work related to quantum computing, we are concerned with the average mixing matrix  MˆX, defined byMˆX=limT→∞1T∫0TH(t)∘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 to12n+2(2J+I+T) where T is the permutation matrix that swaps j   and n+1−jn+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 isn−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.)