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arXiv:1412.4309 · Interdisciplinary Information Sciences

Weak Limit Theorem of a Two-phase Quantum Walk with One Defect

Shimpei ENDO, Takako Endo, Norio Konno, Etsuo Segawa, Masato Takei

Published Jan 1, 2016• 16 citations
arXivHTMLPDF

Abstract

We attempt to analyze a one-dimensional space-inhomogeneous quantum walk (QW) with one defect at the origin, which has two different quantum coins in positive and negative parts. We call the QW ``the two-phase QW with one defect'', which we treated concerning localization theorems. The two-phase QW with one defect has been expected to be a mathematical model of topological insulator which is an intense issue both theoretically and experimentally. In this paper, we derive the weak limit theorem describing the ballistic spreading, and as a result, we obtain the mathematical expression of the whole picture of the asymptotic behavior. Our approach is based mainly on the generating function of the weight of the passages. We emphasize that the time-averaged limit measure is symmetric for the origin, however, the weak limit measure is asymmetric, which implies that the weak limit theorem represents the asymmetry of the probability distribution.

DOI
10.4036/iis.2016.r.01

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We investigate ``the Wojcik model" introduced and studied by Wojcik et al. \cite{wojcik}, 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. Three types of measures have 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. \cite{segawa} suggested, the time-averaged limit and stationary measures are closely related to each other for some QWs with one defect. 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 \cite{watanabe}. 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.

Quantum Information and Computation•Jan 1, 2015•6 citations
We attempt to analyze a one-dimensional space-inhomogeneous quantum walk (QW) with one defect at the origin, which has two different quantum coins in positive and negative parts. We call the QW ``the two-phase QW with one defect'', which we treated concerning localization theorems. The two-phase QW with one defect has been expected to be a mathematical model of topological insulator which is an intense issue both theoretically and experimentally. In this paper, we derive the weak limit theorem describing the ballistic spreading, and as a result, we obtain the mathematical expression of the whole picture of the asymptotic behavior. Our approach is based mainly on the generating function of the weight of the passages. We emphasize that the time-averaged limit measure is symmetric for the origin, however, the weak limit measure is asymmetric, which implies that the weak limit theorem represents the asymmetry of the probability distribution.
We treat a position dependent quantum walk (QW) on the line which we assign two different time-evolution operators to positive and negative parts respectively. We call the model “the two-phase QW” here, which has been expected to be a mathematical model of the topological insulator. We obtain the stationary and time-averaged limit measures related to localization for the two-phase QW with one defect. This is the first result on localization for the two-phase QW. The analytical methods are mainly based on the splitted generating function of the solution for the eigenvalue problem, and the generating function of the weight of the passages of the model. In this paper, we call the methods “the splitted generating function method” and “the generating function method”, respectively. The explicit expression of the stationary measure is asymmetric for the origin, and depends on the initial state and the choice of the parameters of the model. On the other hand, the time-averaged limit measure has a starting point symmetry and localization effect heavily depends on the initial state and the parameters of the model. Regardless of the strong effect of the initial state and the parameters, the time-averaged limit measure also suggests that localization can be always observed for our two-phase QW. Furthermore, our results imply that there is an interesting relation between the stationary and time-averaged limit measures when the parameters of the model have specific periodicities, which suggests that there is a possibility that we can analyze localization of the two-phase QW with one defect from the stationary measure.
We study "the Wojcik model" which is a discrete-time quantum walk (QW) with one defect in one dimension, introduced by Wojcik et al.. For the Wojcik model, we give the weak convergence theorem describing the ballistic behavior of the walker in the probability distribution in a rescaled position-space. In our previous studies, we obtained the time-averaged limit and stationary measures concerning localization for the Wojcik model. As a result, we get the mathematical expression of the whole picture of the behavior of the walker for the Wojcik model. Here the coexistence of localization and the ballistic spreading is one of the peculiar properties of one-dimensional QWs with one defect. Due to the coexistence, it has been strongly expected to utilize QWs to quantum search algorithms. In order to derive the weak convergence theorem, we take advantage of the generating function method. We emphasize that the time-averaged limit measure is symmetric for the origin, however, the weight function in the weak limit measure is asymmetric in general, which implies that the weak convergence theorem represents the asymmetry of the probability distribution. Furthermore, the weak limit measure heavily depends on the phase of the defect and initial state of the walker. Comparing with our previous studies, we also show some numerical results of the probability distribution to confirm that our result is relevant mathematically, and consider the effect of changing the phase and initial coin state on the probability distribution, or the ballistic spreading, which is one of the motivations of our study.
We study a position-dependent discrete-time quantum walk (QW) in one dimension, whose time-evolution operator is built up from two coin operators which are distinguished by phase factors from x≥0x\geq0x≥0 and x≤−1x\leq-1x≤−1. We call the QW the {\it complete two-phase QW} to discern from the two-phase QW with one defect\cite{endosan,maman}. Because of its localization properties, the two-phase QWs can be considered as an ideal mathematical model of topological insulators which are novel quantum states of matter characterized by topological invariants. Employing the complete two-phase QW, we present the stationary measure, and two kinds of limit theorems concerning {\it localization} and the {\it ballistic spreading}, which are the characteristic behaviors in the long-time limit of discrete-time QWs in one dimension. As a consequence, we obtain the mathematical expression of the whole picture of the asymptotic behavior of the walker, including dependences on initial states, in the long-time limit. We also clarify relevant symmetries, which are essential for topological insulators, of the complete two-phase QW, and then derive the topological invariant. Having established both mathematical rigorous results and the topological invariant of the complete two-phase QW, we provide solid arguments to understand localization of QWs in term of topological invariant. Furthermore, by applying a concept of {\it topological protections}, we clarify that localization of the two-phase QW with one defect, studied in the previous work\cite{endosan}, can be related to localization of the complete two-phase QW under symmetry preserving perturbations.
We treat quantum walk (QW) on the line whose quantum coin at each vertex tends to the identity as the distance goes to infinity. We obtain a limit theorem that this QW exhibits localization with not an exponential but a ``power-law" decay around the origin and a ``strongly" ballistic spreading called bottom localization in this paper. This limit theorem implies the weak convergence with linear scaling whose density has two delta measures at x=0x=0x=0 (the origin) and x=1x=1x=1 (the bottom) without continuous parts.
We consider a one-dimensional space-inhomogeneous discrete time quantum walk. This model is the Hadamard walk with one defect at the origin which is different from the model introduced by Wojcik et al. [14]. We obtain a stationary measure of the model by solving the eigenvalue problem and an asymptotic behaviour of the return probability by the path counting approach. Moreover, we get the time-averaged limit measure using the space-time generating function method. The measure is symmetric for the origin and independent of the initial coin state at the starting point. So localization depends only on the parameter which determines the model.
We investigate ``the Wojcik model" introduced and studied by Wojcik et al. \cite{wojcik}, 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. Three types of measures have 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. \cite{segawa} suggested, the time-averaged limit and stationary measures are closely related to each other for some QWs with one defect. 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 \cite{watanabe}. 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.