  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.
