Matrix-valued Szegő polynomials and quantum random walks
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 a...
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Published in | Communications on pure and applied mathematics Vol. 63; no. 4; pp. 464 - 507 |
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Main Authors | , , , |
Format | Journal Article |
Language | English |
Published |
Hoboken
Wiley Subscription Services, Inc., A Wiley Company
01.04.2010
Wiley John Wiley and Sons, Limited |
Subjects | |
Online Access | Get full text |
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Summary: | 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. |
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Bibliography: | istex:D0A913457152FD60D9024584C998B89D2DE9FEB7 Ministry of Education and Science - No. project code MTM2005-08648-C02-01 Air Force Office of Scientific Research (AFOSR) - No. Contract FA9550-08-1-0169 ark:/67375/WNG-VN9MMJQF-J Diputación General de Aragón (Spain) - No. Project E-64 National Science Foundation Grant - No. DMS 0603901 ArticleID:CPA20312 Ministry of Science and Innovation - No. project code MTM2008-06689-C02-01 |
ISSN: | 0010-3640 1097-0312 |
DOI: | 10.1002/cpa.20312 |