Quantum walk on orbit spaces

Abstract

Inspired by the covering-space method in path integrals on multiply connected spaces, we here present a universal formula of time-evolution kernels for continuous- and discrete-time quantum walks on orbit spaces. In this note, we focus on the case in which walkers' configuration space is the orbit space $\mathrm{\ensuremath{\Lambda}}/\mathrm{\ensuremath{\Gamma}}$, where $\mathrm{\ensuremath{\Lambda}}$ is an arbitrary lattice and $\mathrm{\ensuremath{\Gamma}}$ is a discrete group whose action on $\mathrm{\ensuremath{\Lambda}}$ has no fixed points. We show that the time-evolution kernel on $\mathrm{\ensuremath{\Lambda}}/\mathrm{\ensuremath{\Gamma}}$ can be written as a weighted sum of time-evolution kernels on $\mathrm{\ensuremath{\Lambda}}$, where the summation is over the orbit of initial point in $\mathrm{\ensuremath{\Lambda}}$ and weight factors are given by a one-dimensional unitary representation of $\mathrm{\ensuremath{\Gamma}}$. Focusing on one dimension, we present a number of examples of the formula. We also present universal formulas of resolvent kernels, canonical density matrices, and unitary representations of arbitrary groups in quantum walks on $\mathrm{\ensuremath{\Lambda}}/\mathrm{\ensuremath{\Gamma}}$, all of which are constructed in exactly the same way as for the time-evolution kernel.