arXiv:1010.3481

Limit distribution with a combination of density functions for a 2-state quantum walk

Takuya Machida

Published Oct 18, 2010• 3 citations

Abstract

We consider 2-state quantum walks (QWs) on the line, which are defined by two matrices. One of the matrices operates the walk in certain intervals. In the usual QWs starting from the origin, localization does not occur at all. However, our walk can be localized around the origin. In this paper, we present some limit distributions for the walk.

DOI: 10.48550/arxiv.1010.3481

Related papers

LIMIT THEOREMS FOR A LOCALIZATION MODEL OF 2-STATE QUANTUM WALKS

Takuya Machida

We consider 2-state quantum walks (QWs) on the line, which are defined by two matrices. One of the matrices operates the walk at only half-time. In the usual QWs, localization does not occur at all. However, our walk can be localized around the origin. In this paper, we present two limit theorems, that is, one is a stationary distribution and the other is a convergence theorem in distribution.

Jan 1, 201012 citations

LIMIT THEOREMS FOR A LOCALIZATION MODEL OF 2-STATE QUANTUM WALKS

TAKUYA MACHIDA

arXiv:1008.1116International Journal of Quantum InformationApr 1, 20116 citations

Limit theorems of a 3-state quantum walk and its application for discrete uniform measures

MachidaTakuya

Quantum Information and ComputationApr 1, 20150 citations

Limit theorems of a 3-state quantum walk and its application for discrete uniform measures

Takuya Machida

We present two long-time limit theorems of a 3-state quantum walk on the line when the walker starts from the origin. One is a limit measure which is obtained from the probability distribution of the walk at a long-time limit, and the other is a convergence in distribution for the walker’s position in a rescaled space by time. In addition, as an application of the walk, we obtain discrete uniform limit measures from the 3-state walk with a delocalized initial state.

arXiv:1404.1522Quantum Information and ComputationApr 1, 201525 citations

A new type of quantum walks based on decomposing quantum states

Chusei Kiumi

In this paper, the 2-state decomposed-type quantum walk (DQW) on a line is introduced as an extension of the 2-state quantum walk (QW). The time evolution of the DQW is defined with two different matrices, one is assigned to a real component, and the other is assigned to an imaginary component of the quantum state. Unlike the ordinary 2-state QWs, localization and the spreading phenomenon can coincide in DQWs. Additionally, a DQW can always be converted to the corresponding 4-state QW with identical probability measures. In other words, a class of 4-state QWs can be realized by DQWs with 2 states. In this work, we reveal that there is a 2-state DQW corresponding to the 4-state Grover walk. Then, we derive the weak limit theorem of the class of DQWs corresponding to 4-state QWs which can be regarded as the generalized Grover walks.

arXiv:2012.00327Quantum Information and ComputationApr 27, 20214 citations

Limit distribution of a quantum walk driven by a CMV matrix

Takuya Machida

Quantum walks with CMV matrices, five-diagonal unitary matrices, started to be studied in 2003 [1]. The spectral analysis for CMV matrices told us that the quantum walks could localize in distribution depending on the Verblunsky parameters of the matrices. In this paper, we work on a quantum walk whose system is manipulated by a CMV matrix with homogeneous Vervlunsky parameters, and present long-time limit distributions. One can understand from the theory that the quantum walk does not localize and how it approximately distributes after the long-time evolution has been executed on the walk.

Mar 17, 20190 citations