Orthogonal polynomials and perfect state transfer.

Abstract

The aim of this review 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 (PST). 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-neighbour interactions using exceptional orthogonal polynomials (XOPs). We also provide a (non-exhaustive) list of related open questions.This article is part of the theme issue 'Numerical analysis, spectral graph theory, orthogonal polynomials and quantum algorithms'.