Single-qubit unitary gates by graph scattering

We consider the effects of plane-wave states scattering off finite graphs as an approach to implementing single-qubit unitary operations within the continuous-time quantum walk framework of universal quantum computation. Four semi-infinite tails are attached at arbitrary points of a given graph, representing the input and output registers of a single qubit. For a range of momentum eigenstates, we enumerate all of the graphs with up to n = 9 vertices for which the scattering implements a single-qubit gate. As n increases, the number of new unitary operations increases exponentially, and for n > 6 the majority correspond to rotations about axes distributed roughly uniformly across the Bloch sphere. Rotations by both rational and irrational multiples of pi are found.