  A discrete-time quantum walk is the quantum analogue of a Markov chain on a
graph. Zhan [J. Algebraic Combin. 53(4):1187-1213, 2020] proposes a model of
discrete-time quantum walk whose transition matrix is given by two reflections,
using the face and vertex incidence relations of a graph embedded in an
orientable surface. We show that the evolution of a general discrete-time
quantum walk that consists of two reflections satisfies a Chebyshev recurrence,
under a projection. For the vertex-face walk, we prove theorems about perfect
state transfer and periodicity and give infinite families of examples where
these occur. We bring together tools from algebraic and topological graph
theory to analyze the evolution of this walk.
