2009
DOI: 10.1002/cpa.20312
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Matrix‐valued Szegő polynomials and quantum random walks

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 nonnegative integers can … Show more

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Cited by 113 publications
(173 citation statements)
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“…4) The appealing five diagonal representation (2.21) of QW on N is also related so called CMV matrices associated with certain orthogonal polynomials on the unit circle, when restricted to l 2 (Z + ). This fact is used in [11] to study certain properties of deterministic QW.…”
Section: Remarksmentioning
confidence: 99%
“…4) The appealing five diagonal representation (2.21) of QW on N is also related so called CMV matrices associated with certain orthogonal polynomials on the unit circle, when restricted to l 2 (Z + ). This fact is used in [11] to study certain properties of deterministic QW.…”
Section: Remarksmentioning
confidence: 99%
“…1. We consider a Hilbert space H = Z ⊗ C 2 with basis vectors of the form |n ⊗|↑ , |n ⊗|↓ for n ∈ Z.…”
Section: B Quantum Walks and CMV Operatorsmentioning
confidence: 99%
“…This way, a weight of the graph G becomes a positive function defined on the edges of the specified graph G. Below we develop the general theory, illustrate its applications; and we obtain Shannon's result as a special case. An especially attractive statistical mechanics application is [24], and [9]. Now Shannon's view is motivated by signal processing, i.e., engineering of signals, see e.g., [12]: interpolation of functions (signals) on a continuum, determining band-limited functions defined on a continuum from their discrete samples.…”
Section: Signalsmentioning
confidence: 99%