We present an in-depth analytic study of discrete-time quantum walks drivenby a non-reflective coin. Specifically, we compare the properties of thewidely-used Grover coin ${\cal C}_{G}$ that is unitary and reflective (${\calC}_{G}^{2}=\mathbb{I}$) with those of a $3\times3$ "rotational" coin ${\calC}_{60}$ that is unitary but non-reflective (${\calC}_{60}^{2}\not=\mathbb{I}$) and satisfies instead ${\calC}_{60}^{6}=\mathbb{I}$, which corresponds to a rotation by $60^{\circ}$. Whilesuch a modification apparently changes the real-space renormalization group(RG) treatment, we show that nonetheless this non-reflective quantum walkremains in the same universality class as the Grover walk. We first demonstratethe procedure with ${\cal C}_{60}$ for a 3-state quantum walk on aone-dimensional (\emph{1d}) line, where we can solve the RG-recursions inclosed form, in the process providing exact solutions for some difficultnon-linear recursions. Then, we apply the procedure to a quantum walk on a dualSierpinski gasket (DSG), for which we reproduce ultimately the same resultsfound for ${\cal C}_{G}$, further demonstrating the robustness of theuniversality class.
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