arXiv:1010.5295

Quantum Random Walks of Waves

Tianli Feng, Yong-Sheng Zhang, Guangming Zhao, Sheng Liu, Guang‐Can Guo

Published Oct 26, 2010• 1 citation

Abstract

The extremely fascinating behaviors of the quantum walks of particles, which differ much from the classical counterparts, have attracted many physicists. Here we investigate another interesting part of the quantum walks, that is the quantum walks of waves. Firstly, we show the behaviors of the quantum walks of plane wave, which are largely different from the counterparts of either the classical or the quantum walks of particles. Two situations - with and without intermediate time measurements of the walks are considered. At last, it is shown that the quantum walks of plane wave can be used to calculate the evolution of the general wave packets, e.g., Gaussian wave packet.

DOI: 10.48550/arxiv.1010.5295

Related papers

Quantum Walks in Time

Andreas Schreiber

Quantum walks are the quantum mechanical analog to classical random walks. However, due to the wavelike nature of quantum particles, they reveal drastically different dynamics compared to their classical counterpart. These unique properties make quantum walks a new promising platform for quantum algorithms and a powerful tool to understand complex transport phenomena. This thesis presents three experiments specifically designed to send photons on a quantum walk. The experiments are based on a fiber network spreading the photonic wave packets over discrete time bins. With these Quantum Walks in Time we simulate the propagation of quantum particles in various different environments, revealing diverse dynamics such as a fast ballistic spread and an Anderson-like localization effect. Furthermore we use our concept to implement the first quantum walk on a two-dimensional lattice. The underlying topology allows for investigations of the intriguing dynamics of two strongly interacting quantum particles, which reveals unique effects such as the creation of bound states.

Jan 1, 20142 citations

Quantum Walks

Takuya Machida

Quantum walks are quantum counterparts of random walks. While probability distributions of random walks diffusively spread out as the walkers are updating, quantum walks have ballistic behavior. Some of the ballistic behaviors have been revealed in long-time limit theorems and their probability distributions are all far away from the Gaussian distributions, which are known as limit distributions of random walks. In this chapter, we are going to be seeing limit distributions for a standard quantum walk on the line and two kinds of time-dependent quantum walk on the line.

InTech eBooksAug 24, 20161 citation

Quantum walks with random phase shifts

Jozef Košík, Vladimír Bužek, Mark S. Hillery

We investigate quantum walks in multiple dimensions with different quantum coins. We augment the model by assuming that at each step the amplitudes of the coin state are multiplied by random phases. This model enables us to study in detail the role of decoherence in quantum walks and to investigate the quantum-to-classical transition. We also provide classical analog of the quantum random walks studied. Interestingly enough, it turns out that the classical counterparts of some quantum random walks are classical random walks with a memory and biased coin. In addition random phase shifts ``simplify'' the dynamics (the cross-interference terms of different paths vanish on average) and enable us to give a compact formula for the dispersion of such walks.

arXiv:quant-ph/0607092Physical Review AAug 9, 200670 citations

How Quantum is a "Quantum Walk"?

Fereshte Shahbeigi, Seyed Javad Akhtarshenas, A. T. Rezakhani

We characterize quantumness of the so-called quantum walks (whose dynamics is governed by quantum mechanics) by introducing two computable measures which are stronger than the variance of the walker's position probability distribution. The first measure is based on comparing probability distributions of a quantum walk and all classical random walks (through the classical relative entropy of the distributions), and it quantifies non-Gaussianity of the probability distribution of the walk. Next, after assigning a density matrix to classical walks, we introduce a more powerful measure by employing quantum relative entropy. We show that this measure exceeds the first one by the quantum coherence of the walk. There are walks labeled classical by the variance whereas our measures identify some quantumness therein. As an application, we study a model of quantum (energy) transport on a simple lattice, and compare the behavior of its relative transport efficiencies with that of the quantumness. Our measures help partly explain why in some quantum transport phenomena a considerably high efficiency may appear---this is where quantumness is appreciable.

arXiv:1802.07027Feb 20, 20182 citations

General wave packets in a homogeneous field

Lin Qiong

The centroid and width of a general wave packet moving in a homogeneous field in one dimension are obtained as explicit functions of time.The centroid moves like a classical particle,and the width increases with time quadratically.However,the width of the corresponding wave packet in the momentum space does not vary with time,so a plane wave remains to be a plane wave at any time.The time dependence of a Gaussian wave packet is worked out explicitly and the general results are verified.

College PhysicsJan 1, 20030 citations

QUANTUM RANDOM WALK ON FRACTALS

Kai Zhao

Quantum walks are the quantum mechanical analogue of classical random walks. Discrete-time quantum walks have been introduced and studied mostly on the line Z or higher dimensional space Z d but rarely defined on graphs with fractal dimensions because the coin operator depends on the position and the Fourier transform on the fractals is not defined. Inspired by its nature of classical walks, different quantum walks will be defined by choosing different shift and coin operators. When the coin operator is uniform, the results of classical walks will be obtained upon measurement at each step. Moreover, with measurement at each step, our results reveal more information about the classical random walks. In this dissertation, two graphs with fractal dimensions will be considered. The first one is Sierpinski gasket, a degree-4 regular graph with Hausdorff di- mension of df = ln 3/ ln 2. The second is the Cantor graph derived like Cantor set, with Hausdorff dimension of df = ln 2/ ln 3. The definitions and amplitude functions of the quantum walks will be introduced. The main part of this dissertation is to derive a recursive formula to compute the amplitude Green function. The exiting probability will be computed and compared with the classical results. When the generation of graphs goes to infinity, the recursion of the walks will be investigated and the convergence rates will be obtained and compared with the classical counterparts.

Jan 1, 20180 citations