  The Szegedy quantum walk is a discrete time quantum walk model which defines
a quantum analogue of any Markov chain. The long-term behavior of the quantum
walk can be encoded in a matrix called the average mixing matrix, whose columns
give the limiting probability distribution of the walk given an initial state.
We define a version of the average mixing matrix of the Szegedy quantum walk
which allows us to more readily compare the limiting behavior to that of the
chain it quantizes. We prove a formula for our mixing matrix in terms of the
spectral decomposition of the Markov chain and show a relationship with the
mixing matrix of a continuous quantum walk on the chain. In particular, we
prove that average uniform mixing in the continuous walk implies average
uniform mixing in the Szegedy walk. We conclude by giving examples of Markov
chains of arbitrarily large size which admit average uniform mixing in both the
continuous and Szegedy quantum walk.
