Quantum walks in low dimension

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Abstract

Discrete-time quantum walks are defined as a non-commutative analogue of the usual random walks on standard lattices and have been formulated in computer sciences. They are new objects in mathematics and are investigated in various areas, such as computer sciences, quantum physics, probability theory, and discrete geometric analysis. In this article, recent works on point-wise asymptotic behavior and an effective formula for nth power of the discrete-time quantum walks in one dimension are surveyed. The idea to obtain the formula for the nth power in one dimension is applied in this paper to compute the nth power of certain two-dimensional quantum walk, called the Grover walk to obtain a new formula for the two-dimensional Grover walk. The formula for nth power in one dimension has been used to prove a weak limit theorem. In this paper, the large deviation asymptotics, in one dimension, is deduced by using this formula which is a new proof of a previously obtained result.

Original languageEnglish
Title of host publicationGeometric Methods in Physics - 34th Workshop
EditorsPiotr Kielanowski, S. Twareque Ali, Pierre Bieliavsky, Anatol Odzijewicz, Martin Schlichenmaier, Theodore Voronov
PublisherSpringer International Publishing
Pages261-278
Number of pages18
ISBN (Print)9783319317557
DOIs
Publication statusPublished - 2016
Event34th Workshop on Geometric Methods in Physics, 2015 - >Białowieża, Poland
Duration: 2015 Jun 282015 Jul 4

Publication series

NameTrends in Mathematics
Volume0
ISSN (Print)2297-0215
ISSN (Electronic)2297-024X

Other

Other34th Workshop on Geometric Methods in Physics, 2015
CountryPoland
City>Białowieża
Period15/6/2815/7/4

Keywords

  • Asymptotics
  • One-dimensional quantum walks
  • Semi-direct products
  • Two-dimensional Grover walk

ASJC Scopus subject areas

  • Mathematics(all)

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  • Cite this

    Tate, T. (2016). Quantum walks in low dimension. In P. Kielanowski, S. T. Ali, P. Bieliavsky, A. Odzijewicz, M. Schlichenmaier, & T. Voronov (Eds.), Geometric Methods in Physics - 34th Workshop (pp. 261-278). (Trends in Mathematics; Vol. 0). Springer International Publishing. https://doi.org/10.1007/978-3-319-31756-4_21