Matrix-valued Szegő polynomials and quantum random walks

• Published in: Communications on pure and applied mathematics Vol. 63; no. 4; pp. 464 - 507
• Main Authors: Cantero, M. J., Moral, L., Grünbaum, F. Alberto, Velázquez, L.
• Format: Journal Article
• Language: English
• Published: Hoboken Wiley Subscription Services, Inc., A Wiley Company 01.04.2010; Wiley; John Wiley and Sons, Limited
• Subjects: Exact sciences and technology; Fourier analysis; Mathematical analysis; Mathematics; Matrix; Polynomials; Quantum theory; Random walk theory; Sciences and techniques of general use; Special functions; Case study; Integer; Recurrence relation; Random walk; Orthogonal polynomial
• ISSN: 0010-3640; 1097-0312
• DOI: 10.1002/cpa.20312

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 nonnegative 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 nonnegative 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 self‐contained, but we refrain 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, 9]. See also the recent notes [20]. © 2009 Wiley Periodicals, Inc.