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Renormalization of discrete-time quantum walks with a non-Grover coin

We present an in-depth analytic study of discrete-time quantum walks driven by a non-reflective coin. Specifically, we compare the properties of the widely-used Grover coin CG that is unitary and reflective (CG2=I) with those of a 3×3 ‘rotational’ coin C60 that is unitary but non-reflective (C602≠I) and satisfies instead C606=I, which corresponds to a rotation by 60∘. While such a modification apparently changes the real-space renormalization group (RG) treatment, we show that nonetheless this non-reflective quantum walk remains in the same universality class as the Grover walk. We first demonstrate the procedure with C60 for a three-state quantum walk on a one-dimensional (1d) line, where we can solve the RG-recursions in closed form in a process providing exact solutions for some difficult non-linear recursions. Then, we apply the procedure to a quantum walk on a dual Sierpinski gasket (DSG), for which we ultimately reproduce the same results found for CG, further demonstrating the robustness of the universality class.