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Acta Sinica Quantum Optica

Two-dimensional Quantum Random Walk and its Physical Realization

Peng Xue

Published Jan 1, 2011• 0 citations
OpenAlex

Abstract

Quantum random walks have been much studied recently,largely due to their highly non-classical behaviors.In this paper,we review the classical and quantum random walks of one-dimensional,and analyze two classical two-dimensional models and compare them.Furthermore we propose a two-dimensional quantum random walk algorithm.We compute the standard deviation,thereby proving our two-dimensional algorithm is superior to that of one-dimension.We propose a scheme,which is based on single photon with pairs of qubits,to implement it.

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Quantum random walks have been much studied recently,largely due to their highly non-classical behaviors.In this paper,we review the classical and quantum random walks of one-dimensional,and analyze two classical two-dimensional models and compare them.Furthermore we propose a two-dimensional quantum random walk algorithm.We compute the standard deviation,thereby proving our two-dimensional algorithm is superior to that of one-dimension.We propose a scheme,which is based on single photon with pairs of qubits,to implement it.
Quantum walks are interesting models with potential applications to quantum algorithms and physical processes such as photosynthesis. In this paper, we study two models of one-dimensional quantum walks, namely, quantum walks with a moving absorbing wall and quantum walks with one stationary and one moving absorbing wall. For the former, we calculate numerically the survival probability, the rate of change of average position, and the rate of change of standard deviation of the particle's position in the long time limit for different wall velocities. Moreover, we also study the asymptotic behavior and the dependence of the survival probability on the initial particle's state. While for the latter, we compute the absorption probability of the right stationary wall for different velocities and initial positions of the left wall boundary. The results for these two models are compared with those obtained for the classical model. The difference between the results obtained for the quantum and classical models can be attributed to the difference in the probability distributions.
Quantum random walks have attracted special interest because they could lead to new quantum algorithms. Photons can carry orbital angular momentum (OAM) thereby offering a practical realization of a high-dimensional quantum information carrier. By employing OAM of photons, we experimentally realized the one-dimensional discrete-time quantum random walk. Three steps of a one-dimensional quantum random walk were implemented in our protocol showing the obvious difference between quantum and classical random walks.
The effect of unitary noise on the discrete one-dimensional quantum walk is studied using computer simulations. For the noiseless quantum walk, starting at the origin (n=0)(n=0)(n=0) at time t=0,t=0,t=0, the position distribution Pt(n){P}_{t}(n)Pt​(n) at time t is very different from the Gaussian distribution obtained for the classical random walk. Furthermore, its standard deviation, $\ensuremath{\sigma}(t)$ scales as $\ensuremath{\sigma}(t)\ensuremath{\sim}t,$ unlike the classical random walk for which $\ensuremath{\sigma}(t)\ensuremath{\sim}\sqrt{t}.$ It is shown that when the quantum walk is exposed to unitary noise, it exhibits a crossover from quantum behavior for short times to classical-like behavior for long times. The crossover time is found to be $T\ensuremath{\sim}{\ensuremath{\alpha}}^{\ensuremath{-}2},$ where $\ensuremath{\alpha}$ is the standard deviation of the noise.
Quantum walks, the quantum mechanical counterpart of classical random walks, is an advanced tool for building quantum algorithms that has been recently shown to constitute a universal model of quantum computation. Quantum walks is now a solid field of research of quantum computation full of exciting open problems for physicists, computer scientists and engineers. In this paper we review theoretical advances on the foundations of both discrete- and continuous-time quantum walks, together with the role that randomness plays in quantum walks, the connections between the mathematical models of coined discrete quantum walks and continuous quantum walks, the quantumness of quantum walks, a summary of papers published on discrete quantum walks and entangle-ment as well as a succinct review of experimental proposals and realizations of discrete-time quantum walks. Furthermore, we have reviewed several algorithms based on both discrete- and continuous-time quantum walks as well as a most important result: the computational universality of both continuous- and discrete-time quantum walks.
A random walk is known as a random process which describes a path including a succession of random steps in the mathematical space. It has increasingly been popular in various disciplines such as mathematics and computer science. Furthermore, in quantum mechanics, quantum walks can be regarded as quantum analogues of classical random walks. Classical random walks and quantum walks can be used to calculate the proximity between nodes and extract the topology in the network. Various random walk related models can be applied in different fields, which is of great significance to downstream tasks such as link prediction, recommendation, computer vision, semi-supervised learning, and network embedding. In this article, we aim to provide a comprehensive review of classical random walks and quantum walks. We first review the knowledge of classical random walks and quantum walks, including basic concepts and some typical algorithms. We also compare the algorithms based on quantum walks and classical random walks from the perspective of time complexity. Then we introduce their applications in the field of computer science. Finally we discuss the open issues from the perspectives of efficiency, main-memory volume, and computing time of existing algorithms. This study aims to contribute to this growing area of research by exploring random walks and quantum walks together.
A quantum walk is the quantum analog of the classical random walks. Despite their simple structure they form a universal platform to implement any algorithm of quantum computation. However, it is very hard to realize quantum walks with a sufficient number of iterations in quantum systems due to their sensitivity to environmental influences and subsequent loss of coherence. Here we present a scalable implementation scheme for one-dimensional quantum walks for arbitrary number of steps using the orbital angular momentum modes of classical light beams. Furthermore, we show that using the same setup with a minor adjustment we can also realize electric quantum walks.