  We consider the definition of quantum walks on directed graphs. Call a
directed graph reversible if, for each pair of vertices (i, j), if i is
connected to j then there is a path from j to i. We show that reversibility is
a necessary and sufficient condition for a directed graph to allow the notion
of a discrete-time quantum walk, and discuss some implications of this
condition. We present a method for defining a "partially quantum" walk on
directed graphs that are not reversible.
