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Journal of the Physical Society of Japan

Nonunitarity of Simple 3+1-Dimensional Quantum Walk Models Corresponding to the Weyl and Dirac Equations

TAKUO YAMAMOTO, Shinji Yamashita, Kohei Eguchi

Published Jan 25, 2013• 0 citations
Publisher

Abstract

It is well known that \(1+1\)-dimensional Weyl and Dirac equations in the relativistic quantum mechanics are related to some quantum walks. However, in \(3+1\)-dimensions, discrete-time quantum walks corresponding to the above two equations have not been obtained. We show that the \(3+1\)-dimensional Weyl and Dirac walkers cannot be obtained by generalizing the above well-known model to \(3+1\)-dimensional cases. Apparently, these \(3+1\)-dimensional extended models seem to be well defined, but they do not have unitary time evolution. Thus, the square of the magnitude of each spinor is not able to be any probability distribution. In addition, we present two Schrödinger walkers, and one of which has unitary time evolution. This unitary Schrödinger walker can move to all sites at each time step, unlike the existing models. Therefore, it is mentioned that adequate models of \(3+1\)-dimensional Weyl and Dirac walkers will also be given like this unitary Schrödinger walker.

DOI
10.7566/jpsj.82.024002

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It is well known that (1+1)-dimensional Weyl and Dirac equations in the relativistic quantum mechanics are related to some quantum walks. However, in (3+1)-dimensions, discrete-time quantum walks corresponding to the above two equations have not been obtained. We show that the (3+1)-dimensional Weyl and Dirac walkers cannot be obtained by generalizing the above well-known model to (3+1)-dimensional cases. Apparently, these (3+1)-dimensional extended models seem to be well defined, but they do not have unitary time evolution. Thus, the square of the magnitude of each spinor is not able to be any probability distribution. In addition, we present two Schrödinger walkers, and one of which has unitary time evolution. This unitary Schrödinger walker can move to all sites at each time step, unlike the existing models. Therefore, it is mentioned that adequate models of (3+1)-dimensional Weyl and Dirac walkers will also be given like this unitary Schrödinger walker.
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.
A quantum walk describes the discrete unitary evolution of a quantum particle on a discrete graph. Some quantum walks, referred to as the Weyl and Dirac walks, provide a description of the free evolution of relativistic quantum fields in the small wave-vector regime. The clash between the intrinsic discreteness of quantum walks and the continuous symmetries of special relativity is resolved by giving a definition of change of inertial frame in terms of a change of values of the constants of motion, which leaves the walk operator unchanged. Starting from the family of 1 + 1 dimensional Dirac walks with all possible values of the mass parameter, we introduce a unique walk encompassing the latter as an extra degree of freedom, and we derive its group of changes of inertial frames. This symmetry group contains a non-linear realization of SO+(2,1)⋉R3; since one of the two space-like dimensions does not correspond to an actual spatial degree of freedom but rather the mass, we interpret it as a 2 + 1 dimensional de Sitter group. This group also contains a non-linear realization of the proper orthochronous Poincaré group SO+(1,1)⋉R2 in 1 + 1 dimension, as the ones considered within the framework of doubly special relativity, which recovers the usual relativistic symmetry in the limit of small wave-vectors and masses. Surprisingly, for the Dirac walk with a fixed value of the mass parameter, the group of allowed changes of reference frame does not have a consistent interpretation in the limit of small wave-vectors.
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.
The time-evolution equation of a one-dimensional quantum walker is exactly mapped to the three-dimensional Weyl equation for a zero-mass particle with spin 1∕21∕21∕2, in which each wave number kkk of the walker's wave function is mapped to a point q(k)\mathbf{q}(k)q(k) in the three-dimensional momentum space and q(k)\mathbf{q}(k)q(k) makes a planar orbit as kkk changes its value in $[\ensuremath{-}\ensuremath{\pi},\ensuremath{\pi})$. The integration over kkk providing the real-space wave function for a quantum walker corresponds to considering an orbital state of a Weyl particle, which is defined as a superposition (curvilinear integration) of the energy-momentum eigenstates of a free Weyl equation along the orbit. Konno's novel distribution function of a quantum walker's pseudovelocities in the long-time limit is fully controlled by the shape of the orbit and how the orbit is embedded in the three-dimensional momentum space. The family of orbital states can be regarded as a geometrical representation of the unitary group U(2) and the present study will propose a new group-theoretical point of view for quantum-walk problems.
Discrete-time quantum walks (QWs) are transportation models of single quantum particles over a lattice. Their evolution is driven through causal and local unitary operators. QWs are a powerful tool for quantum simulation of fundamental physics, as some of them have a continuum limit converging to well-known physics partial differential equations, such as the Dirac or the Schr\"odinger equation. In this paper, we show how to recover the Dirac equation in (3+1)(3+1)(3+1) dimensions with a QW evolving in a tetrahedral space. This paves the way to simulate the Dirac equation on a curved space-time. This also suggests an ordered scheme for propagating matter over a spin network, of interest in loop quantum gravity, where matter propagation has remained an open problem.
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.