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The (standard) average mixing matrix of a continuous-time quantum walk is
computed by taking the expected value of the mixing matrices of the walk under
the uniform sampling distribution on the real line. In this paper we consider
alternative probability distributions, either discrete or continuous, and first
we show that several algebraic properties that hold for the average mixing
matrix still stand for this more general setting. Then, we provide examples of
graphs and choices of distributions where the average mixing matrix behaves in
an unexpected way: for instance, we show that there are probability
distributions for which the average mixing matrices of the paths on three or
four vertices have constant entries, opening a significant line of
investigation about how to use classical probability distributions to sample
quantum walks and obtain desired quantum effects. We present results connecting
the trace of the average mixing matrix and quantum walk properties, and we show
that the Gram matrix of average states is the average mixing matrix of a
certain related distribution. Throughout the text, we employ concepts of
classical probability theory not usually seen in texts about quantum walks.
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