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

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.