Quantum Walk on Orbit Spaces

• URL: http://arxiv.org/abs/2301.03193v2
• Date: Sat, 17 Jun 2023 08:00:00 GMT
• Title: Quantum Walk on Orbit Spaces
• Authors: Satoshi Ohya
• Abstract summary: We present a universal formula of time-evolution kernels for continuous- and discrete-time quantum walks on orbit spaces. We show that the time-evolution kernel on $Lambda/Gamma$ can be written as a weighted sum of time-evolution kernels on $Lambda$. We also present universal formulas of resolvent kernels, canonical density matrices, and unitary representations of arbitrary groups in quantum walks on $Lambda/Gamma$.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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