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 $hatsigma_z$, while for $theta=pi/4$ we obtain the Hadamard gate. The optimal $theta$ sequences present non-trivial patterns, with mostly $thetaapprox 0$ alternated with $thetaapprox pi/4$ 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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