Quantum walks in low dimension

Tatsuya Tate

doi:10.1007/978-3-319-31756-4_21

Quantum walks in low dimension

Tatsuya Tate

SCI - Mathematics

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

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 language	English
Title of host publication	Geometric Methods in Physics - 34th Workshop
Editors	Piotr Kielanowski, S. Twareque Ali, Pierre Bieliavsky, Anatol Odzijewicz, Martin Schlichenmaier, Theodore Voronov
Publisher	Springer International Publishing
Pages	261-278
Number of pages	18
ISBN (Print)	9783319317557
DOIs	https://doi.org/10.1007/978-3-319-31756-4_21
Publication status	Published - 2016
Event	34th Workshop on Geometric Methods in Physics, 2015 - >Białowieża, Poland Duration: 2015 Jun 28 → 2015 Jul 4

Publication series

Name	Trends in Mathematics
Volume	0
ISSN (Print)	2297-0215
ISSN (Electronic)	2297-024X

Conference

Conference	34th Workshop on Geometric Methods in Physics, 2015
Country/Territory	Poland
City	>Białowieża
Period	15/6/28 → 15/7/4

Keywords

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

Access to Document

10.1007/978-3-319-31756-4_21

Cite this

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title = "Quantum walks in low dimension",

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.",

keywords = "Asymptotics, One-dimensional quantum walks, Semi-direct products, Two-dimensional Grover walk",

author = "Tatsuya Tate",

note = "Publisher Copyright: {\textcopyright} 2016 Springer International Publishing.; 34th Workshop on Geometric Methods in Physics, 2015 ; Conference date: 28-06-2015 Through 04-07-2015",

year = "2016",

doi = "10.1007/978-3-319-31756-4_21",

language = "English",

isbn = "9783319317557",

series = "Trends in Mathematics",

publisher = "Springer International Publishing",

pages = "261--278",

editor = "Piotr Kielanowski and Ali, {S. Twareque} and Pierre Bieliavsky and Anatol Odzijewicz and Martin Schlichenmaier and Theodore Voronov",

booktitle = "Geometric Methods in Physics - 34th Workshop",

}

Tate, T 2016, Quantum walks in low dimension. in P Kielanowski, ST Ali, P Bieliavsky, A Odzijewicz, M Schlichenmaier & T Voronov (eds), Geometric Methods in Physics - 34th Workshop. Trends in Mathematics, vol. 0, Springer International Publishing, pp. 261-278, 34th Workshop on Geometric Methods in Physics, 2015, >Białowieża, Poland, 15/6/28. https://doi.org/10.1007/978-3-319-31756-4_21

Quantum walks in low dimension. / Tate, Tatsuya.
Geometric Methods in Physics - 34th Workshop. ed. / Piotr Kielanowski; S. Twareque Ali; Pierre Bieliavsky; Anatol Odzijewicz; Martin Schlichenmaier; Theodore Voronov. Springer International Publishing, 2016. p. 261-278 (Trends in Mathematics; Vol. 0).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

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AB - 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.

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ER -

Quantum walks in low dimension

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