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arXiv:1008.5109

Localization of discrete-time quantum walks on a half line via the CGMV method

Norio Konno, Etsuo Segawa

Published Aug 30, 2010• 7 citations
arXivHTMLPDF

Abstract

We study discrete-time quantum walks on a half line by means of spectral analysis. Cantero et al. [1] showed that the CMV matrix, which gives a recurrence relation for the orthogonal Laurent polynomials on the unit circle [2], expresses the dynamics of the quantum walk. Using the CGMV method introduced by them, the name is taken from their initials, we obtain the spectral measure for the quantum walk. As a corollary, we give another proof for localization of the quantum walk on homogeneous trees shown by Chisaki et al. [3].

DOI
10.48550/arxiv.1008.5109

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We study discrete-time quantum walks on a half line by means of spectral analysis. Cantero et al. [1] showed that the CMV matrix, which gives a recurrence relation for the orthogonal Laurent polynomials on the unit circle [2], expresses the dynamics of the quantum walk. Using the CGMV method introduced by them, the name is taken from their initials, we obtain the spectral measure for the quantum walk. As a corollary, we give another proof for localization of the quantum walk on homogeneous trees shown by Chisaki et al. [3].
We study discrete-time quantum walks on a half line by means of spectral analysis. Cantero et al. showed that the CMV matrix, which gives a recurrence relation for the orthogonal Laurent polynomials on the unit circle, expresses the dynamics of the quantum walk. Using the CGMV method introduced by them, the name is taken from their initials, we obtain the spectral measure for the quantum walk. As a corollary, we give another proof for localization of the quantum walk on homogeneous trees shown by Chisaki et al.
Abstract We define quantization scheme for discrete-time random walks on the half-line consistent with Szegedy's quantization of finite Markov chains. Motivated by the Karlin and McGregor description of discrete-time random walks in terms of polynomials orthogonal with respect to a measure with support in the segment [-1,1], we represent the unitary evolution operator of the quantum walk in terms of orthogonal polynomials on the unit circle. We find the relation between transition probabilities of the random walk with the Verblunsky coefficients of the corresponding polynomials of the quantum walk. We show that the both polynomials systems and their measures are connected by the classical Szegö map. Our scheme can be applied to arbitrary Karlin and McGregor random walks and generalizes the so called Cantero--Grünbaum--Moral--Velázquez method. We illustrate our approach on example of random walks related to the Jacobi polynomials. Then we study quantization of random walks with constant transition probabilities where the corresponding polynomials on the unit circle have two-periodic real Verblunsky coefficients. We present geometric construction of the spectrum of such polynomials (in the general complex case) which generalizes the known construction for the Geronimus polynomials. In the Appendix we present the explicit form, in terms of Chebyshev polynomials of the second kind, of polynomials orthogonal on the unit circle and polynomials orthogonal on the real line with coefficients of arbitrary period. 2010 Mathematics Subject Classification. 81P68, 05C81, 42C05, 94A11.
We consider quantum random walks (QRW) on the integers, a subject that has been considered in the last few years in the framework of quantum computation. We show how the theory of CMV matrices gives a natural tool to study these processes and to give results that are analogous to those that Karlin and McGregor developed to study (classical) birth-and-death processes using orthogonal polynomials on the real line. In perfect analogy with the classical case the study of QRWs on the set of non-negative integers can be handled using scalar valued (Laurent) polynomials and a scalar valued measure on the circle. In the case of classical or quantum random walks on the integers one needs to allow for matrix valued versions of these notions. We show how our tools yield results in the well known case of the Hadamard walk, but we go beyond this translation invariant model to analyze examples that are hard to analyze using other methods. More precisely we consider QRWs on the set of non- negative integers. The analysis of these cases leads to phenomena that are absent in the case of QRWs on the integers even if one restricts oneself to a constant coin. This is illustrated here by studying recurrence properties of the walk, but the same method can be used for other purposes. The presentation here aims at being selfcontained, but we re- frain from trying to give an introduction to quantum random walks, a subject well surveyed in the literature we quote. For two excellent reviews, see [1, 18]. See also the recent notes [19].
We consider a discrete-time quantum walk W_{t,\kappa} at time t on a graph with joined half lines J_\kappa, which is composed of \kappa half lines with the same origin. Our analysis is based on a reduction of the walk on a half line. The idea plays an important role to analyze the walks on some class of graphs with symmetric initial states. In this paper, we introduce a quantum walk with an enlarged basis and show that W_{t,\kappa} can be reduced to the walk on a half line even if the initial state is asymmetric. For W_{t,\kappa}, we obtain two types of limit theorems. The first one is an asymptotic behavior of W_{t,\kappa} which corresponds to localization. For some conditions, we find that the asymptotic behavior oscillates. The second one is the weak convergence theorem for W_{t,\kappa}. On each half line, W_{t,\kappa} converges to a density function like the case of the one-dimensional lattice with a scaling order of t. The results contain the cases of quantum walks starting from the general initial state on a half line with the general coin and homogeneous trees with the Grover coin.
Abstract In this paper, we treat quantum walks in a two-dimensional lattice with cutting edges along a straight boundary introduced by Asboth and Edge (2015 Phys. Rev . A 91 022324) in order to study one-dimensional edge states originating from topological phases of matter and to obtain collateral evidence of how a quantum walker reacts to the boundary. Firstly, we connect this model to the CMV matrix, which provides a 5-term recursion relation of the Laurent polynomial associated with spectral measure on the unit circle. Secondly, we explicitly derive the spectra of bulk and edge states of the quantum walk with the boundary using spectral analysis of the CMV matrix. Thirdly, while topological numbers of the model studied so far are well-defined only when gaps in the bulk spectrum exist, we find a new topological number defined only when there are no gaps in the bulk spectrum. We confirm that the existence of the spectrum for edge states derived from the CMV matrix is consistent with the prediction from a bulk-edge correspondence using topological numbers calculated in the cases where gaps in the bulk spectrum do or do not exist. Finally, we show how the edge states contribute to the asymptotic behavior of the quantum walk through limit theorems of the finding probability. Conversely, we also propose a differential equation using this limit distribution whose solution is the underlying edge state.