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 $\theta$, such that for $\theta =0$ we have the ${\widehat{\sigma }}_z$, while for $\theta =\pi /4$ we obtain the Hadamard gate. The optimal $\theta$ sequences present nontrivial patterns, with mostly $\theta \approx 0$ alternating with $\theta \approx \pi /4$ values after increasingly long periods. We provide an analysis of the entanglement properties, quasienergy spectrum, and survival probability, providing a full physical picture.

• Received 20 April 2020
• Accepted 29 June 2020

DOI:https://doi.org/10.1103/PhysRevA.102.022223