Virtually Abelian quantum walks

Abstract

We study discrete-time quantum walks on Cayley graphs of non-Abelian groups, focusing on the easiest case of virtually Abelian groups. We present a technique to reduce the quantum walk to an equivalent one on an Abelian group with coin system having larger dimension. This method allows one to extend the notion of wave-vector to the virtually Abelian case and study analytically the walk dynamics. We apply the technique in the case of two quantum walks on virtually Abelian groups with planar Cayley graphs, finding the exact solution in terms of dispersion relation.