  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.
