Unveiling two-dimensional discrete quantum walks dynamics via dispersion relations

The discrete, or coined, quantum walk (QW) [1] is a process originally introduced as the quantum counterpart of the classical random walk (RW). In both cases there is a walker and a coin: at every time step the coin is tossed and the walker moves depending on the toss output. Unlike the RW, in the QW the walker and coin are quantum in nature what allows the coherent superpositions right/left and head/tail happen. This feature endows the QW with outstanding properties, such as making the standard deviation of the position of an initially localized walker grow linearly with time t, unlike the RW in which this growth goes as t1/2. This has strong consequences in algorithmics and is one of the …