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CATEGORIES:Isaac Newton Institute Seminar Series
SUMMARY:Quantum random walks and orthogonal polynomials -
Grunbaum\, FA (UC\, Berkeley)
DTSTART;TZID=Europe/London:20090323T140000
DTEND;TZID=Europe/London:20090323T150000
UID:TALK17525AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/17525
DESCRIPTION:This is joint work with M.J. Cantero\, L. Moral an
d L. Velazquez from Zaragoza\, Spain.\nWe consider
quantum random walks (QRW) on the integers\, a su
bject that has been considered\nin the last few ye
ars in the framework of quantum computation.\nWe s
how how the theory of CMV matrices gives a natural
tool to study these processes and\nto give result
s that are analogous to those that Karlin and McGr
egor developed to study (classical)\nbirth-and-dea
th processes using orthogonal polynomials on the r
eal line.\nIn perfect analogy with the classical c
ase the study of QRWs on the set of non-negative i
ntegers\ncan be handled using scalar valued (Laure
nt) polynomials and a scalar valued measure on the
circle.\nIn the case of classical or quantum rand
om walks on the integers one needs to allow for ma
trix\nvalued versions of these notions.\nWe show h
ow 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 a
re hard to analyze using other\nmethods. More prec
isely we consider QRWs on the set of non-negative
integers. The analysis of\nthese cases leads to ph
enomena that are absent in the case of QRWs on the
integers even if one\nrestricts oneself to a cons
tant coin. This is illustrated here by studying re
currence properties of the\nwalk\, but the same me
thod can be used for other purposes.
LOCATION:Seminar Room 1 Newton Institute
CONTACT:Mustapha Amrani
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