Quantum Walk on Orbit Spaces

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.