  Discrete-time quantum walks allow Floquet topological insulator materials to
be explored using controllable systems such as ultracold atoms in optical
lattices. By numerical simulations, we study the robustness of topologically
protected edge states in the presence of decoherence in one- and
two-dimensional discrete-time quantum walks. We also develop a simple
analytical model quantifying the robustness of these edge states against either
spin or spatial dephasing, predicting an exponential decay of the population of
topologically protected edge states. Moreover, we present an experimental
proposal based on neutral atoms in spin-dependent optical lattices to realize
spatial boundaries between distinct topological phases. Our proposal relies on
a new scheme to implement spin-dependent discrete shift operations in a
two-dimensional optical lattice. We analyze under realistic decoherence
conditions the experimental feasibility of observing unidirectional,
dissipationless transport of matter waves along boundaries separating distinct
topological domains.
