Matrix valued Szego polynomials and quantum random walks

TL;DR

This paper introduces matrix-valued Szego polynomials as a tool to analyze quantum random walks on integers, extending classical orthogonal polynomial methods to quantum settings and revealing new phenomena in recurrence properties.

Contribution

It develops a framework using CMV matrices and matrix-valued orthogonal polynomials to study quantum random walks on non-negative integers, surpassing previous scalar approaches.

Findings

Analysis of recurrence properties in quantum walks on non-negative integers.

Application of matrix-valued Szego polynomials to quantum walk models.

Extension of classical birth-and-death process techniques to quantum case.

Abstract

We consider quantum random walks (QRW) on the integers, a subject that has been considered in the last few years in the framework of quantum computation. We show how the theory of CMV matrices gives a natural tool to study these processes and to give results that are analogous to those that Karlin and McGregor developed to study (classical) birth-and-death processes using orthogonal polynomials on the real line. In perfect analogy with the classical case the study of QRWs on the set of non-negative integers can be handled using scalar valued (Laurent) polynomials and a scalar valued measure on the circle. In the case of classical or quantum random walks on the integers one needs to allow for matrix valued versions of these notions. We show how our tools yield results in the well known case of the Hadamard walk, but we go beyond this translation invariant model to analyze examples that are hard to analyze using other methods. More precisely we consider QRWs on the set of non- negative integers. The analysis of these cases leads to phenomena that are absent in the case of QRWs on the integers even if one restricts oneself to a constant coin. This is illustrated here by studying recurrence properties of the walk, but the same method can be used for other purposes. The presentation here aims at being selfcontained, but we re- frain from trying to give an introduction to quantum random walks, a subject well surveyed in the literature we quote. For two excellent reviews, see [1, 18]. See also the recent notes [19].