Renormalization of Discrete-Time Quantum Walks with non-Grover Coins

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 ${\cal C}_{G}$ that is unitary and reflective (${\cal C}_{G}^{2}=\mathbb{I}$) with those of a $3\times3$ "rotational" coin ${\cal C}_{60}$ that is unitary but non-reflective (${\cal C}_{60}^{2}\not=\mathbb{I}$) and satisfies instead ${\cal C}_{60}^{6}=\mathbb{I}$, which corresponds to a rotation by $60^{\circ}$. 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 ${\cal C}_{60}$ for a 3-state quantum walk on a one-dimensional (\emph{1d}) line, where we can solve the RG-recursions in closed form, in the 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 reproduce ultimately the same results found for ${\cal C}_{G}$, further demonstrating the robustness of the universality class.