Understanding and controlling N-dimensional quantum walks via dispersion relations. Application to the 2D and 3D Grover walks: Diabolical points and more

The discrete quantum walk in N dimensions is analyzed from the perspective of its dispersion relations. This allows understanding known properties, as well as designing new ones when spatially extended initial conditions are considered. This is done by deriving wave equations in the continuum, which are generically of the Schr"odinger type, and allow devising interesting behaviors, such as ballistic propagation without deformation, or the generation of almost flat probability distributions, what is corroborated numerically. There are however special points where the energy surfaces display intersections and, near them, the dynamics is entirely different. Applications to the two- and three-dimensional Grover walks are presented.