Quantum walks on directed graphs
  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. 
Ashley Montanaro