Renormalization of discrete-time quantum walks with a non-Grover coin
Stefan Boettcher, Joshua L. Pughe-Sanford · Journal of Statistical Mechanics Theory and Experiment · 2018
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 C G that is unitary and reflective ( C G 2 = I ) with those of a 3 × 3 ‘rotational’ coin C 60 that is unitary but non-reflective ( C 60 2 ≠ I ) and satisfies instead C 60 6 = 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 C 60 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 C G