Orthogonal Polynomials and Perfect State Transfer

TL;DR

This review explores how orthogonal polynomials are applied in quantum information processing, particularly in analyzing perfect state transfer in quantum walks and extending classical methods to quantum systems.

Contribution

It provides a comprehensive introduction to orthogonal polynomials in quantum walks, highlighting their role in detecting perfect state transfer and extending classical results to quantum contexts.

Findings

Orthogonal polynomials can detect perfect state transfer in quantum walks.

Extensions to quantum walks with non-nearest neighbor interactions are possible using exceptional orthogonal polynomials.

The paper discusses open questions in the application of orthogonal polynomials to quantum information.

Abstract

The aim of this review paper is to discuss some applications of orthogonal polynomials in quantum information processing. The hope is to keep the paper self contained so that someone wanting a brief introduction to the theory of orthogonal polynomials and continuous time quantum walks on graphs may find it in one place. In particular, we focus on the associated Jacobi operators and discuss how these can be used to detect perfect state transfer. We also discuss how orthogonal polynomials have been used to give results which are analogous to those given by Karlin and McGregor when studying classical birth and death processes. Finally, we show how these ideas have been extended to quantum walks with more than nearest neighbor interactions using exceptional orthogonal polynomials. We also provide a (non-exhaustive) list of related open questions.