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Quantum walks are promising for information processing tasks because onregular graphs they spread quadratically faster than random walks. Staticdisorder, however, can turn the tables: unlike random walks, quantum walks cansuffer Anderson localization, whereby the spread of the walker stays within afinite region even in the infinite time limit. It is therefore important tounderstand when we can expect a quantum walk to be Anderson localized and whenwe can expect it to spread to infinity even in the presence of disorder. Inthis work we analyze the response of a generic one-dimensional quantum walk --the split-step walk -- to different forms of static disorder. We find thatintroducing static, symmetry-preserving disorder in the parameters of the walkleads to Anderson localization. In the completely disordered limit, however, adelocalization sets in, and the walk spreads subdiffusively. Using an efficientnumerical algorithm, we calculate the bulk topological invariants of thedisordered walk, and interpret the disorder-induced Anderson localization anddelocalization transitions using these invariants.

Localization, delocalization, and topological phase transitions in the one-dimensional split-step quantum walk