Optimizing the spatial spread of a quantum walk

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.