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arXiv:quant-ph/0607092 · Physical Review A

Quantum walks with random phase shifts

Jozef Košík, Vladimír Bužek, Mark S. Hillery

Published Aug 9, 2006• 70 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physreva.74.022310

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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.
We look at two possible routes to classical behavior for the discrete quantum random walk on the integers: decoherence in the quantum "coin" which drives the walk, or the use of higher-dimensional (or multiple) coins to dilute the effects of interference. We use the position variance as an indicator of classical behavior and find analytical expressions for this in the long-time limit; we see that the multicoin walk retains the "quantum" quadratic growth of the variance except in the limit of a new coin for every step, while the walk with decoherence exhibits "classical" linear growth of the variance even for weak decoherence.
Quantum random walks have been much studied recently, largely due to their highly nonclassical behavior. In this paper, we study one possible route to classical behavior for the discrete quantum random walk on the line: the use of multiple quantum ``coins'' (or more generally, coins of higher dimension) in order to diminish the effects of interference between paths. We find solutions to this system in terms of the single-coin random walk, and compare the asymptotic limit of these solutions to numerical simulations. We find exact analytical expressions for the time dependence of the first two moments, and show that in the long-time limit the ``quantum-mechanical'' behavior of the one-coin walk persists, even if each coin is flipped only twice. We further show that this is generic for a very broad class of possible walks, and that this behavior disappears only in the limit of a new coin for every step of the walk.
Quantum walk (QW), which is considered as the quantum counterpart of the classical random walk (CRW), is actually the quantum extension of CRW from the single-coin interpretation. The sequential unitary evolution engenders correlation between different steps in QW and leads to a non-binomial position distribution. In this paper, we propose an alternative quantum extension of CRW from the ensemble interpretation, named quantum random walk (QRW), where the walker has many unrelated coins, modeled as two-level systems, initially prepared in the same state. We calculate the walker’s position distribution in QRW for different initial coin states with the coin operator chosen as Hadamard matrix. In one-dimensional case, the walker’s position is the asymmetric binomial distribution. We further demonstrate that in QRW, coherence leads the walker to perform directional movement. For an initially decoherenced coin state, the walker’s position distribution is exactly the same as that of CRW. Moreover, we study QRW in 2D lattice, where the coherence plays a more diversified role in the walker’s position distribution.
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.