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arXiv:1305.1722 · Quantum Information and Computation

One-dimensional quantum walks via generating function and the CGMV method

Norio Konno, Etsuo Segawa

Published Oct 1, 2014• 7 citations
arXivHTMLPDF

Abstract

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 $x=0$ (the origin) and $x=1$ (the bottom) without continuous parts.

DOI
10.26421/qic14.13-14-8

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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 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 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.
Abstract In this paper, we study the properties of lackadaisical quantum walks (LQWs) on a line. This model was first proposed in Wong (2015 J. Phys. A: Math. Theor . 48 435304) as a quantum analogue of lazy random walks where each vertex is attached to τ self-loops. We derive an analytic expression for the localization probability of the walker at the origin after infinite steps, and obtain the peak velocities of the walker. We also calculate the wave function of the walker starting from the origin and obtain a long-time approximation for the entire probability density function. As an application of the density function, we prove that lackadaisical quantum walks spread ballistically for arbitrary τ , and give an analytic solution for the variance of the walker’s probability distribution.
We study a discrete-time quantum walk (QW) on the line with a single phase at the origin which was introduced and studied by Wojcik et al.[1]. We call the model "Wojcik model" here. Konno et al.[2] investigated other types of QWs with one defect at the origin. They presented a method which gives the stationary measure corresponding to localization for the QWs by use of the generating functions splitted in positive and negative parts respectively. In this paper, we call the method "the splitted generating function method (the SGF method)". To clarify in detail which QW is appropriate for the method in the case of study is one of the important challenges to investigate localization properties for various QWs. As for the Wojcik model, we solve the eigenvalue problem by the SGF method and our results agree with Ref.[1]. From the solution of the problem, we derive a stationary measure with an exponential decay for the position. The explicit expression for the stationary measure is symmetric for the origin and ensures localization depending on the initial coin state.
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.