  We investigate one-dimensional (1D) discrete time quantum walks (QWs) with
spatially or temporally random defects as a consequence of interactions with
random environments. We focus on the QWs with chiral symmetry in a topological
phase, and reveal that chiral symmetry together with bipartite nature of the
QWs brings about intriguing behaviors such as coexistence of topologically
protected edge states at zero energy and Anderson transitions in the 1D chiral
class at non-zero energy in their dynamics. Contrary to the previous studies,
therefore, the spatially disordered QWs can avoid complete localization due to
the Anderson transition. It is further confirmed that the edge states are
robust for spatial disorder but not for temporal disorder.
