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SUMMARY:Quantum random walks and orthogonal polynomials - Grunbaum\, FA (U
 C\, Berkeley)
DTSTART:20090323T140000Z
DTEND:20090323T150000Z
UID:TALK17525@talks.cam.ac.uk
CONTACT:Mustapha Amrani
DESCRIPTION:This is joint work with M.J. Cantero\, L. Moral and L. Velazqu
 ez from Zaragoza\, Spain.\nWe consider quantum random walks (QRW) on the i
 ntegers\, a subject that has been considered\nin the last few years in the
  framework of quantum computation.\nWe show how the theory of CMV matrices
  gives a natural tool to study these processes and\nto give results that a
 re analogous to those that Karlin and McGregor developed to study (classic
 al)\nbirth-and-death processes using orthogonal polynomials on the real li
 ne.\nIn perfect analogy with the classical case the study of QRWs on the s
 et of non-negative integers\ncan be handled using scalar valued (Laurent) 
 polynomials and a scalar valued measure on the circle.\nIn the case of cla
 ssical or quantum random walks on the integers one needs to allow for matr
 ix\nvalued versions of these notions.\nWe show how our tools yield results
  in the well known case of the Hadamard walk\, but we go\nbeyond this tran
 slation invariant model to analyze examples that are hard to analyze using
  other\nmethods. More precisely we consider QRWs on the set of non-negativ
 e integers. The analysis of\nthese cases leads to phenomena that are absen
 t in the case of QRWs on the integers even if one\nrestricts oneself to a 
 constant coin. This is illustrated here by studying recurrence properties 
 of the\nwalk\, but the same method can be used for other purposes.
LOCATION:Seminar Room 1 Newton Institute
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