Renormalization group for quantum walks

Abstract

We present a detailed introduction to the discrete-time quantum walk problem, in close analogy with the classical ordinary and persistent random walk. This approach facilitates a uniform application of the renormalization group that highlights similarities and differences between the classical and the quantum walk problem. Specifically, we discuss the renormalization group treatment for the mean-square displacement of a walker starting from a single site on the 1d-line for ordinary and persistent random walks and the quantum walk. We outline the significance of universality for quantum walks and the control this might provide for quantum algorithms. We use our RG method to verify that all 2-state quantum walks on the 1d-line are in the same universality class.