TY - GEN

T1 - Quantum walks in low dimension

AU - Tate, Tatsuya

N1 - Funding Information:
The author is partially supported by JSPS Grants-in-Aid for Scientific Research (No. 25400068).
Publisher Copyright:
© 2016 Springer International Publishing.
Copyright:
Copyright 2020 Elsevier B.V., All rights reserved.

PY - 2016

Y1 - 2016

N2 - 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.

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.

KW - Asymptotics

KW - One-dimensional quantum walks

KW - Semi-direct products

KW - Two-dimensional Grover walk

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U2 - 10.1007/978-3-319-31756-4_21

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

M3 - Conference contribution

AN - SCOPUS:84982943382

SN - 9783319317557

T3 - Trends in Mathematics

SP - 261

EP - 278

BT - Geometric Methods in Physics - 34th Workshop

A2 - Kielanowski, Piotr

A2 - Ali, S. Twareque

A2 - Bieliavsky, Pierre

A2 - Odzijewicz, Anatol

A2 - Schlichenmaier, Martin

A2 - Voronov, Theodore

PB - Springer International Publishing

T2 - 34th Workshop on Geometric Methods in Physics, 2015

Y2 - 28 June 2015 through 4 July 2015

ER -