  Quantum walks are promising for information processing tasks because on
regular graphs they spread quadratically faster than random walks. Static
disorder, however, can turn the tables: unlike random walks, quantum walks can
suffer Anderson localization, whereby the spread of the walker stays within a
finite region even in the infinite time limit. It is therefore important to
understand when we can expect a quantum walk to be Anderson localized and when
we can expect it to spread to infinity even in the presence of disorder. In
this work we analyze the response of a generic one-dimensional quantum walk --
the split-step walk -- to different forms of static disorder. We find that
introducing static, symmetry-preserving disorder in the parameters of the walk
leads to Anderson localization. In the completely disordered limit, however, a
delocalization sets in, and the walk spreads subdiffusively. Using an efficient
numerical algorithm, we calculate the bulk topological invariants of the
disordered walk, and interpret the disorder-induced Anderson localization and
delocalization transitions using these invariants.
