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Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE

Analytic views of quantum walks

Ian Fuss, Lang White, Peter J. Sherman, Sanjeev Naguleswaran

Published Jun 7, 2007• 0 citations
Publisher

Abstract

The first general analytic solutions for the one-dimensional quantum walk in position and momentum space are derived. These solutions reveal new symmetry features of quantum walk probability densities and insight into the behaviour of their moments. The analytic expressions for the quantum walk probability distributions provide a means of modelling quantum phenomena that is analogous to that provided by random walks in the classical domain.

DOI
10.1117/12.726362

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The first general analytic solutions for the one-dimensional quantum walk in position and momentum space are derived. These solutions reveal new symmetry features of quantum walk probability densities and insight into the behaviour of their moments. The analytic expressions for the quantum walk probability distributions provide a means of modelling quantum phenomena that is analogous to that provided by random walks in the classical domain.
The behavior of one-dimensional quantum random walks is strikingly different from that of classical ones. However, when decoherence is involved, the limiting distributions take on many classical features over time. In this paper, we study the decoherence on both position and ``coin'' spaces of the particle. We propose an analytical approach to investigate these phenomena and obtain the generating functions which encode all the features of these walks. Specifically, from these generating functions, we find exact analytic expressions of several moments for the time and noise dependence of position. Moreover, the limiting position distributions of decoherent quantum random walks are shown to be Gaussian in an analytical manner. These results explicitly describe the relationship between the system and the level of decoherence.
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.
In this paper we address the quantum random walk on the real line. Specifically, we utilize a dynamical system formulation of the walk, which leads to a momentum space expression of the probability amplitude that is a function of only the initial condition. Our focus is, for the most part, limited to the Hadamard walk. As such, this closed form expression is not as general as that given by [1]. This lack of generality is offset by the ease with which we obtain the expression, and the insight offered by it. Our closed form expression allows us to easily compute the walk pdf, hence the cdf (cumulative distribution function). It is shown that the cdf converges to its limiting form relatively quickly. We push the simple mathematics in an to attempt to obtain a closed form expression for this form. But it becomes too involved to take it to completion in this paper, without running the risk of losing appreciation for the simplicity of our approach.
We define and analyze quantum computational variants of \nrandom walks on one-dimensional lattices. In particular, we analyze a quantum analog of the symmetric random \nwalk, which we call the Hadamard walk. Several striking \ndifferences between the quantum and classical cases are ob- served. For example, when unrestricted in either direction, \nthe Hadamard walk has position that is nearly uniformly \ndistributed in the range [-t/√2, t/√2] after t steps, which \nis in sharp contrast to the classical random walk, which has \ndistance O(√t) from the origin with high probability. With \nan absorbing boundary immediately to the left of the starting position, the probability that the walk exits to the left is 2/π, and with an additional absorbing boundary at location n, the probability that the walk exits to the left actually increases, approaching 1/√2 in the limit. In the classical case both values are 1.