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arXiv:1703.05890 · Physical Review A

Simple derivation of the Weyl and Dirac quantum cellular automata

Philippe Raynal

Published Jun 30, 2017• 8 citations
arXivHTMLPDF

Abstract

We consider quantum cellular automata on a body-centered cubic lattice and provide a simple derivation of the only two homogenous, local, isotropic, and unitary two-dimensional automata [G. M. D'Ariano and P. Perinotti, Phys. Rev. A 90, 062106 (2014)]. Our derivation relies on the notion of Gram matrix and emphasizes the link between the transition matrices that characterize the automata and the body-centered cubic lattice: The transition matrices essentially are the matrix representation of the vertices of the lattice's primitive cell. As expected, the dynamics of these two automata reduce to the Weyl equation in the limit of small wave vectors and continuous time. We also briefly examine the four-dimensional case, where we find two one-parameter families of automata that reduce to the Dirac equation in a suitable limit.

DOI
10.1103/physreva.95.062344

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We consider quantum cellular automata on a body-centered cubic lattice and provide a simple derivation of the only two homogenous, local, isotropic, and unitary two-dimensional automata [G. M. D'Ariano and P. Perinotti, Phys. Rev. A 90, 062106 (2014)]. Our derivation relies on the notion of Gram matrix and emphasizes the link between the transition matrices that characterize the automata and the body-centered cubic lattice: The transition matrices essentially are the matrix representation of the vertices of the lattice's primitive cell. As expected, the dynamics of these two automata reduce to the Weyl equation in the limit of small wave vectors and continuous time. We also briefly examine the four-dimensional case, where we find two one-parameter families of automata that reduce to the Dirac equation in a suitable limit.
The thesis focuses on the analytical and the numerical study of the Weyl and Dirac Quantum Cellular Automata (QCAs). A QCA describes the unitary local evolution of a denumerable collection of quantum systems in mutual interaction. The Weyl QCA can be derived under the assumptions of linearity, unitarity, locality, homogeneity and isotropy and one can show that it recovers the relativistic evolution of the Weyl equation in the limit of small wave-vectors. Under these assumptions a QCA can be described in terms of a set of so-called transition matrices. In the specific case of the Weyl automaton the transition matrices generate a semigroup allowing for the explicit computation of the propagator in position space as a sum over paths. The computation of the propagator is simplified by encoding the lattice paths as binary strings, so that the problems reduces to a combinatorial problem for the binary strings. The thesis also addresses the Thirring QCA, featuring an on-site interaction. We provide a starting point for the study of the perturbation theory in this case.
We define quantum cellular automata as infinite quantum lattice systems with discrete time dynamics, such that the time step commutes with lattice translations and has strictly finite propagation speed. In contrast to earlier definitions this allows us to give an explicit characterization of all local rules generating such automata. The same local rules also generate the global time step for automata with periodic boundary conditions. Our main structure theorem asserts that any quantum cellular automaton is structurally reversible, i.e., that it can be obtained by applying two blockwise unitary operations in a generalized Margolus partitioning scheme. This implies that, in contrast to the classical case, the inverse of a nearest neighbor quantum cellular automaton is again a nearest neighbor automaton. We present several construction methods for quantum cellular automata, based on unitaries commuting with their translates, on the quantization of (arbitrary) reversible classical cellular automata, on quantum circuits, and on Clifford transformations with respect to a description of the single cells by finite Weyl systems. Moreover, we indicate how quantum random walks can be considered as special cases of cellular automata, namely by restricting a quantum lattice gas automaton with local particle number conservation to the single particle sector.
We show that the Dirac quantum cellular automaton [A. Bisio, G. M. D'Ariano, and A. Tosini, Ann. Phys. (N.Y.) 354, 244 (2015)] shares many properties in common with the discrete-time quantum walk. These similarities can be exploited to study the automaton as a unitary process that takes place at regular time steps on a one-dimensional lattice, in the spirit of general quantum cellular automata. In this way, it becomes an alternative to the quantum walk, with a dispersion relation that can be controlled by a parameter that plays a similar role to the coin angle in the quantum walk. The Dirac Hamiltonian is recovered under a suitable limit. We provide two independent analytical approximations to the long-term probability distribution. It is shown that, starting from localized conditions, the asymptotic value of the entropy of entanglement between internal and motional degrees of freedom overcomes the known limit that is approached by the quantum walk for the same initial conditions and is similar to the ones achieved by highly localized states of the Dirac equation.
In recent work [quant-ph/0405174] by Schumacher and Werner was discussed an abstract algebraic approach to a model of reversible quantum cellular automata (CA) on a lattice. It was used special model of CA based on partitioning scheme and so there is a question about quantum CA derived from more general, standard model of classical CA. In present work is considered an approach to definition of a scheme with "history", valid for quantization both irreversible and reversible classical CA directly using local transition rules. It is used language of vectors in Hilbert spaces instead of C*-algebras, but results may be compared in some cases. Finally, the quantum lattice gases, quantum walk and "bots" are also discussed briefly.
Quantum walks on lattices can give rise to one-particle relativistic wave equations in the long-wavelength limit. In going to multiple particles, quantum cellular automata (QCA) are natural generalizations of quantum walks. In one spatial dimension, the quantum walk can be ``promoted'' to a QCA that, in the long-wavelength limit, gives rise to the Dirac quantum field theory (QFT) for noninteracting fermions. This QCA/QFT correspondence has both theoretical and practical applications, but there are obstacles to similar constructions in two or more spatial dimensions. Here we show that a method of construction employing distinguishable particles confined to the completely antisymmetric subspace yields a QCA in two spatial dimensions that gives rise to the two-dimensional (2D) Dirac QFT. Generalizing to 3D will entail some additional complications, but no conceptual barriers. We examine how this construction evades the ``no-go'' results in earlier work.
This thesis presents a model of Quantum Cellular Automata (QCA). The presented formalism is a natural quantization of the classical Cellular Automata (CA). It is based on a lattice of qudits, and an update rule consisting of local unitary operators that commute with their own lattice translations. One purpose of this model is to act as a theoretical model of quantum computation, similar to the quantum circuit model. The main advantage that QCA have over quantum circuits is that QCA make considerably fewer demands on the underlying hardware. In particular, as opposed to direct implementations of quantum circuits, the global evolution of the lattice in the QCA model does not assume independent control over individual \emph{qudits}. Rather, all qudits are to be addressed collectively in parallel. The QCA model is also shown to be an appropriate abstraction for space-homogeneous quantum phenomena, such as quantum lattice gases, spin chains and others. Some results that show the benefits of basing the model on local unitary operators are shown: computational universality, strong connections to the circuit model, simple implementation on quantum hardware, and a series of applications. A detailed discussion will be given on one particular application of QCA that lies outside either computation or simulation: single-spin measurement. This algorithm uses the techniques developed in this thesis to achieve a result normally considered hard in physics. It serves well as an example of why QCA are interesting in their own right.