Limit Theorems for the Disordered Quantum Walk

We study the disordered quantum walk in one dimension, and obtain the weak limit theorem.Comment: Contains an open proble

Spectral analysis of discrete-time quantum walks in the quarter plane

Using the Cantero-Grunbaum-Moral-Velazquez (CGMV) method, we obtain the spectral measure for the quantum walk.Comment: Contains an open problem showing the relationship between quantum walks in the quarter plane and quantum walks on homogeneous tree

Limit Theorems For the Grover Walk Without Memory

We consider the Grover walk as a 4-state quantum walk without memory in one dimension. The walker in our 4-state quantum walk moves to the left or right. We compute the stationary distribution of the walk, in addition, we obtain the weak limit theoremComment: 15 pages, contains an extreme open proble

Stationary measure for three-state quantum walk

We focus on the three-state quantum walk(QW) in one dimension. In this paper, we give the stationary measure in general condition, originated from the eigenvalue problem. Firstly, we get the transfer matrices by our new recipe, and solve the eigenvalue problem. Then we obtain the general form of the stationary measure for concrete initial state and eigenvalue. We also show some specific examples of the stationary measure for the three-state QW. One of the interesting and crucial future problems is to make clear the whole picture of the set of stationary measures.Comment: 12 page

Localization of an inhomogeneous discrete-time quantum walk on the line

We investigate a space-inhomogeneous discrete-time quantum walk in one dimension. We show that the walk exhibits localization by a path counting method.Comment: 10 pages, 1 figure, minor corrections, Journal-ref added

Sojourn times of the Hadamard walk in one dimension

The Hadamard walk is a typical model of the discrete-time quantum walk. We investigate sojourn times of the Hadamard walk on a line by a path counting method.Comment: 14 pages, title changed, minor corrections, Quantum Information Processing (in press

Limit Theorems for the Fibonacci Quantum Walk

We study the discrete-time quantum walk in one-dimension governed by the Fibonacci transformation .We show localization does not occur for the Fibonacci quantum walk by investigating the stationary distribution of the walk, in addition, we obtain the weak limit theorem.Comment: Contains an extreme proble

Weak limit theorem for a one-dimensional split-step quantum walk

This paper proves a weak limit theorem for a one-dimensional split-step quantum walk and investigates the limit density function. In the density function, the difference between two Konno's functions appears

Limit measures of inhomogeneous discrete-time quantum walks in one dimension

We treat three types of measures of the quantum walk (QW) with the spatial perturbation at the origin, which was introduced by [1]: time averaged limit measure, weak limit measure, and stationary measure. From the first two measures, we see a coexistence of the ballistic and localized behaviors in the walk as a sequential result following [1,2]. We propose a universality class of QWs with respect to weak limit measure. It is shown that typical spatial homogeneous QWs with ballistic spreading belong to the universality class. We find that the walk treated here with one defect also belongs to the class. We mainly consider the walk starting from the origin. However when we remove this restriction, we obtain a stationary measure of the walk. As a consequence, by choosing parameters in the stationary measure, we get the uniform measure as a stationary measure of the Hadamard walk and a time averaged limit measure of the walk with one defect respectively.Comment: 14 pages, minor corrections, Quantum Information Processing (in press

The time-averaged limit measure of the Wojcik model

We investigate "the Wojcik model" introduced and studied by Wojcik et al., which is a one-defect quantum walk (QW) having a single phase at the origin. They reported that giving a phase at one point causes an astonishing effect for localization. There are three types of measures having important roles in the study of QWs: time-averaged limit measure, weak limit measure, and stationary measure. The first two measures imply a coexistence of localized behavior and the ballistic spreading in the QW. As Konno et al. suggested, the time-averaged limit and stationary measures are closely related to each other for some models. In this paper, we focus on a relation between the two measures for the Wojcik model. The stationary measure was already obtained by our previous work. Here, we get the time-averaged limit measure by several methods. Our results show that the stationary measure is a special case of the time-averaged limit measure.Comment: 28 pages, the name was revised from Watanabe to Endo for marriage, the order of author's names was also changed, the affiliation of the author was also changed, the notation of the second reference was also change