Optimizing the spatial spread of a quantum walk

Abstract

We devise a protocol to build 1D time-dependent quantum walks in 1D maximizing the spatial spread throughout the procedure. We allow only one of the physical parameters of the coin-tossing operator to vary, i.e. the angle $\theta$, such that for $\theta =0$ we have the ${\hat{\sigma }}_z$, while for $\theta =\pi /4$ we obtain the Hadamard gate. The optimal $\theta$ sequences present non-trivial patterns, with mostly $\theta \approx 0$ alternated with $\theta \approx \pi /4$ values after increasingly long periods. We provide an analysis of the entanglement properties, quasi-energy spectrum and survival probability, providing a full physical picture.

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