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

Abstract

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 ${\mathcal{C}}_G$ that is unitary and reflective (${\mathcal{C}}_G^2=\mathbb{I}$) with those of a $3\times 3$ "rotational" coin ${\mathcal{C}}_{60}$ that is unitary but non-reflective (${\mathcal{C}}_{60}^2\ne \mathbb{I}$) and satisfies instead ${\mathcal{C}}_{60}^6=\mathbb{I}$, 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 ${\mathcal{C}}_{60}$ 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 ${\mathcal{C}}_G$, further demonstrating the robustness of the universality class.

arXiv comment14 pages, RevTex4, related information can be found at http://www.physics.emory.edu/faculty/boettcher/

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Cited by in corpus (1)

• An algorithm to factorize quantum walks into shift and coin operations