  Discrete quantum walks are periodically driven systems with discrete time
evolution. In contrast to ordinary Floquet systems, no microscopic Hamiltonian
exists, and the one-period time evolution is given directly by a series of
unitary operators. Regarding each constituent unitary operator as a discrete
time step, we formulate discrete space-time symmetry in quantum walks and
evaluate the corresponding symmetry protected topological phases. In
particular, we study chiral and/or time-glide symmetric topological quantum
walks in this formalism. Due to discrete nature of time evolution,the
topological classification is found to be different from that in conventional
Floquet systems. As a concrete example, we study a two-dimensional quantum walk
having both chiral and time-glide symmetries, and identify the anomalous edge
states protected by these symmetries.
