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arXiv:quant-ph/0508096 · Physical Review A

Relativistic quantum walks

Frederick W. Strauch

Published May 23, 2006• 115 citations
arXivHTMLPDF

Abstract

By pursuing the deep relation between the one-dimensional Dirac equation and quantum walks, the physical role of quantum interference in the latter is explained. It is shown that the time evolution of the probability density of a quantum walker, initially localized on a lattice, is directly analogous to relativistic wave-packet spreading. Analytic wave-packet solutions reveal a striking connection between the discrete and continuous-time quantum walks.

DOI
10.1103/physreva.73.054302

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By pursuing the deep relation between the one-dimensional Dirac equation and quantum walks, the physical role of quantum interference in the latter is explained. It is shown that the time evolution of the probability density of a quantum walker, initially localized on a lattice, is directly analogous to relativistic wave-packet spreading. Analytic wave-packet solutions reveal a striking connection between the discrete and continuous-time quantum walks.
The mathematical relationship between the discrete-time and continuous-time quantum walks and the one-dimensional Dirac equation is explored by studying a class of solutions for each, expressed in terms of the generalized, regular, and modified Bessel functions, respectively. Rigorous limits connecting these solutions are established. In addition, new analytical and numerical results are presented for quantum walks and the Dirac equation, including entanglement, relativistic localization and wave packet spreading, and normal and anomalous Zitterbewegung.
Quantum walk models have been used as an algorithmic tool for quantum computation and to describe various physical processes. This article revisits the relationship between relativistic quantum mechanics and the quantum walks. We show the similarities of the mathematical structure of the decoupled and coupled forms of the discrete-time quantum walk to that of the Klein-Gordon and Dirac equations, respectively. In the latter case, the coin emerges as an analog of the spinor degree of freedom. Discrete-time quantum walk as a coupled form of the continuous-time quantum walk is also shown by transforming the decoupled form of the discrete-time quantum walk to the Schr\"odinger form. By showing the coin to be a means to make the walk reversible and that the Dirac-like structure is a consequence of the coin use, our work suggests that the relativistic causal structure is a consequence of conservation of information. However, decoherence (modeled by projective measurements on position space) generates entropy that increases with time, making the walk irreversible and thereby producing an arrow of time. The Lieb-Robinson bound is used to highlight the causal structure of the quantum walk to put in perspective the relativistic structure of the quantum walk, the maximum speed of walk propagation, and earlier findings related to the finite spread of the walk probability distribution. We also present a two-dimensional quantum walk model on a two-state system to which the study can be extended.
Discrete-time quantum walks are considered a discretization of Dirac equations and the study for them has been getting attention since around 2000. In this paper, we focus on a quantum walk on a line related to a Weyl equation in a space. The quantum walker with two inner states, also interpreted as a quantum particle with the down-spin and the up-spin state in physics, spreads on the line and we analyze the chance of finding the walker at each position after it carries out a unitary evolution a lot of times. The result is reported as a long-time limit distribution from which one can see an approximation to the finding probability.
The Dirac equation can be modelled as a quantum walk (QW), whose main features are being: discrete in time and space (i.e. a unitary evolution of the wave-function of a particle on a lattice); homogeneous (i.e. translation-invariant and time-independent) and causal (i.e. information propagates at a bounded speed, in a strict sense). This link, which was proposed already by Succi and Benzi, Bialynicki-Birula and Meyer, is shown to hold for Bargmann–Wigner equations and symmetric hyperbolic systems in general. We then analytically prove the convergence of the solution of the QW to the solution of the Cauchy problem for the Dirac equation. We do so by adapting a powerful method from standard numerical analysis, which is of general interest to the field of quantum simulation. At the practical level, this result provides precise error bounds and convergence rates, thereby validating the QW as a quantum simulation scheme. At the theoretical level, it reinforces the status of this QW as a simple, discrete toy model of relativistic particles.
It is known that any positive-energy state of a free Dirac particle that is initially highly localized evolves in time by spreading at speeds close to the speed of light. As recently indicated by Strauch, this general phenomenon, and the resulting ``two-horned'' distributions of position probability along any axis through the point of initial localization, can be interpreted in terms of a quantum random walk, in which the roles of ``coin'' and ``walker'' are naturally associated with the spin and translational degrees of freedom in a discretized version of Dirac's equation. We investigate the relationship between these two evolutions analytically and show how the evolved probability density on the xxx axis for the Dirac particle at any time ttt can be obtained from the asymptotic form of the probability distribution for the position of a ``quantum walker.'' The case of a highly localized initial state is discussed as an example.
A continuous-time quantum walk of a phase-modulated wave packet is considered in a one-dimensional equispacing lattice {| k >| k ∈ Z }. The interaction ℏc/Δx\hbar c/\Delta xℏc/Δx between adjacent sites drives the packet unidirectionally, where Δ x denotes the lattice spacing. If the initial state is spread over N consecutive sites and phase-modulated by exp (i k θ), the packet travels ballistically with a velocity v = (1 - N -1 ) c cos θ.