Optimizing the spatial spread of a quantum walk
Gonzalo Martín-Vázquez, Javier Rodríguez-Laguna
Abstract
We devise a protocol to build one-dimensional time-dependent quantum walks, 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 $\ensuremath{\theta}$, such that for $\ensuremath{\theta}=0$ we have the ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\sigma}}}_{z}$, while for $\ensuremath{\theta}=\ensuremath{\pi}/4$ we obtain the Hadamard gate. The optimal $\ensuremath{\theta}$ sequences present nontrivial patterns, with mostly $\ensuremath{\theta}\ensuremath{\approx}0$ alternating with $\ensuremath{\theta}\ensuremath{\approx}\ensuremath{\pi}/4$ values after increasingly long periods. We provide an analysis of the entanglement properties, quasienergy spectrum, and survival probability, providing a full physical picture.