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Russian-Korean International Symposium on Science and Technology (KORUS)

Anderson localization in 1-D quantum random walk

A. N. Bondarenko, Дедок Василий Александрович

Published Oct 4, 2006• 2 citations
Publisher

Abstract

In this paper we intend to present some new results concerning two quantum models: quantum scattering model and discrete quantum random walk. We present the analytical solution of the 1D quantum random walk and give an explicit formula of the quantum return probability. The exact solution provides weak Anderson localization in discrete quantum random walk theory. Connectivity between these theories is introduced.

DOI
10.1109/korus.2005.1507636

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In this paper we intend to present some new results concerning two quantum models: quantum scattering model and discrete quantum random walk. We present the analytical solution of the 1D quantum random walk and give an explicit formula of the quantum return probability. The exact solution provides weak Anderson localization in discrete quantum random walk theory. Connectivity between these theories is introduced.
Abstract We present an overview of two models of quantum random walk. In the first model, the discrete quantum random walk, we present the explicit solution for the recurring amplitude of the quantum random walk on a one-dimensional lattice. We also introduce a new method of solving the problem of random walk in the most general case and use it to derive the hitting amplitude for quantum random walk on the hypercube. The second is a special model based on a local interaction between neighboring spin-1/2 particles on a one-dimensional lattice. We present explicit results for the relevant quantities and obtain an upper bound on the speed of convergence to limiting probability distribution.
We study the evolution of a system performing a one-dimensional quantum walk in the presence of static phase disorder. The same model also describes the propagation of classical light pulses in photonic mesh lattices. We study the interplay between the coupling (i.e. the bias of the "quantum coin") and disorder. We provide an exact analytical expression for the localization length for two limiting cases of strong and weak phase disorder. In all the cases of interest we supply numerical simulations for participation ratio, Lyapunov exponent and the return probability as functions of the coupling parameter.
A theoretical study of the phenomenon of Anderson localization in a generalized discrete time quantum walk will be reported. The generalized discrete time quantum walk is controlled by a quantum coin unitary matrix on each step of particle motion. The quantum coin matrix depends on four angles that can be engineered with microwave pulses in qubit chains. In the absence of disorder in quantum coins parameters the quantum walk dynamics is determined by two-bands dispersion relationship which becomes completely flat in the limit of vanishing kinetic energy. Various types of disorder result in the localization of quantum particle. In particular, kinetic energy disorder leads to logarithmic divergence of the localization length at spectral symmetry points. By making use of the transfer matrix technique we were able to obtain analytical expressions for the frequency dependent localization length in two limits, i.e. weak and strong disorder, and we anticipate that it will be useful for various applications.
A return probability of random walks is one of the interesting subjects. As it is well known, the return probability strongly depends on the structure of the space where the random walker moves. On the other hand, the return probability of quantum walks, which are quantum models corresponding to random walks, has also been investigated to some extend lately. In this paper, we take care of a discrete-time three-state quantum walk on a hexagonal lattice from the view point of mathematics. The mathematical result shows a limit of the return probability when the walker starts off at the origin. The result of the limit tells us about a possibility of localization at the position and a dependence of localization on the initial state.
Discrete-time quantum walks, quantum generalizations of classical random walks, provide a framework for quantum information processing, quantum algorithms, and quantum simulation of condensed-matter systems. The key property of quantum walks, which lies at the heart of their quantum information applications, is the possibility for a parametric quantum speedup in propagation compared to classical random walks. In this work we study propagation of quantum walks on percolation-generated two-dimensional random lattices. In large-scale simulations of topological and trivial split-step walks, we identify distinct prediffusive and diffusive behaviors at different timescales. Importantly, we show that even arbitrarily weak concentrations of randomly removed lattice sites give rise to a complete breakdown of the superdiffusive quantum speedup, reducing the motion to ordinary diffusion. By increasing the randomness, quantum walks eventually stop spreading due to Anderson localization. Near the localization threshold, we find that the quantum walks become subdiffusive. The fragility of quantum speedup implies dramatic limitations for quantum information applications of quantum walks on random geometries and graphs.
Recently Mc Gettrick introduced and studied a discrete-time 2-state quantum walk (QW) with a memory in one dimension. He gave an expression for the amplitude of the QW by path counting method. Moreover he showed that the return probability of the walk is more than 1/2 for any even time. In this paper, we compute the stationary distribution by considering the walk as a 4-state QW without memory. Our result is consistent with his claim. In addition, we obtain the weak limit theorem of the rescaled QW. This behavior is strikingly different from the corresponding classical random walk and the usual 2-state QW without memory as his numerical simulations suggested.