
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 , where is an arbitrary lattice and is a discrete group whose action on has no fixed points. We show that the time-evolution kernel on can be written as a weighted sum of time-evolution kernels on , where the summation is over the orbit of initial point in and weight factors are given by a one-dimensional unitary representation of . 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…
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Taxonomy
TopicsQuantum Computing Algorithms and Architecture · Quantum-Dot Cellular Automata · Quantum and electron transport phenomena
