Renormalization and Scaling in Quantum Walks

We show how to extract the scaling behavior of quantum walks using the renormalization group (RG). We introduce the method by efficiently reproducing well-known results on the one-dimensional lattice. As a nontrivial model, we apply this method to the dual Sierpinski gasket and obtain its exact, closed system of RG-recursions. Numerical iteration suggests that under rescaling the system length, $L^{\mathrm{\prime }}=2L$, characteristic times rescale as $t^{\mathrm{\prime }}=2^{d_w}t$ with the exact walk exponent $d_w={\log }_2\sqrt{5}=1.1609\dots$. Despite the lack of translational invariance, this is very close to the ballistic spreading, $d_w=1$, found for regular lattices. However, we argue that an extended interpretation of the traditional RG formalism will be needed to obtain scaling exponents analytically. Direct simulations confirm our RG-prediction for $d_w$ and furthermore reveal an immensely rich phenomenology for the spreading of the quantum walk on the gasket. Invariably, quantum interference localizes the walk completely with a site-access probability that declines with a powerlaw from the initial site, in contrast with a classical random walk, which would pass all sites with certainty.