Quantum Walk on Orbit Spaces

TL;DR

This paper derives a universal formula for quantum walk kernels on orbit spaces, expressing them as sums over group actions, and provides examples and extensions to resolvent kernels and density matrices.

Contribution

It introduces a universal formula for quantum walk kernels on orbit spaces, generalizing the covering-space method to both continuous and discrete cases.

Findings

Kernel expressed as sum over group orbits with unitary weights

Universal formulas for resolvent kernels and density matrices

Examples provided for one-dimensional orbit spaces

Abstract

Inspired by the covering-space method in path integral 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 $\Lambda/\Gamma$, where $\Lambda$ is an arbitrary lattice and $\Gamma$ is a discrete group whose action on $\Lambda$ has no fixed points. We show that the time-evolution kernel on $\Lambda/\Gamma$ can be written as a weighted sum of time-evolution kernels on $\Lambda$, where the summation is over the orbit of initial point in $\Lambda$ and weight factors are given by a one-dimensional unitary representation of $\Gamma$. 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 $\Lambda/\Gamma$, all of which are constructed in exactly the same way as for the time-evolution kernel.