Disordered quantum walks in two-dimensional lattices

The properties of the two-dimensional quantum walk with point, line, and circle disorders in phase are reported. Localization is observed in the two-dimensional quantum walk with certain phase disorder and specific initial coin states. We give an explanation of the localization behavior via the localized stationary states of the unitary operator of the walker + coin system and the overlap between the initial state of the whole system and the localized stationary states.