Perfect state transfer in quantum walks on orientable maps

TL;DR

This paper investigates discrete-time quantum walks on orientable maps, demonstrating conditions for perfect state transfer and periodicity, and introduces algebraic and topological methods for analyzing their evolution.

Contribution

It establishes new theoretical results on perfect state transfer and periodicity in quantum walks on orientable maps, combining algebraic and topological graph theory tools.

Findings

Proves Chebyshev recurrence for quantum walk evolution

Identifies conditions for perfect state transfer and periodicity

Provides infinite families of examples exhibiting these phenomena

Abstract

A discrete-time quantum walk is the quantum analogue of a Markov chain on a graph. Zhan [J. Algebraic Combin. 53(4):1187-1213, 2020] proposes a model of discrete-time quantum walk whose transition matrix is given by two reflections, using the face and vertex incidence relations of a graph embedded in an orientable surface. We show that the evolution of a general discrete-time quantum walk that consists of two reflections satisfies a Chebyshev recurrence, under a projection. For the vertex-face walk, we prove theorems about perfect state transfer and periodicity and give infinite families of examples where these occur. We bring together tools from algebraic and topological graph theory to analyze the evolution of this walk.