  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.

Quantum Physics