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Fundamental theories of physics

The Cellular Automaton

Gerard ’t Hooft

Published Jan 1, 2016• 9 citations
Publisher

Abstract

A time-reversible cellular automaton can most easily be described by regarding its evolution law as a two-step process, for instance by first allowing the even sites to update, then the odd sites. The discrete Hamiltonian operator procedure of Chap. 19 may be used. Subsequently, we apply a perturbative algebra, starting from the Baker Campbell Hausdorff expansion. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

DOI
10.1007/978-3-319-41285-6_21

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We wrap up the main outcomes of our investigations. Even if not all questions and issues could be addressed as precisely and persuasively as one would wish, we do find that the assumption of an underlying theory describing a single, definite world with completely deterministic evolution laws, can be maintained and pursued further. In the more distant future, it is hoped that the Cellular Automaton Theory will enable us to select credible models for grand unification an quantized gravity; in the mean while, the Cellular Automaton Interpretation will be helpful in visualizing what we are actually doing when we are involved with quantum mechanical calculations. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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A time-reversible cellular automaton can most easily be described by regarding its evolution law as a two-step process, for instance by first allowing the even sites to update, then the odd sites. The discrete Hamiltonian operator procedure of Chap. 19 may be used. Subsequently, we apply a perturbative algebra, starting from the Baker Campbell Hausdorff expansion. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
Abstract We present a one dimensional reversible block cellular automaton, where the time evolution is dictated by a period 3 cycle of update rules. At each time step a subset of the cells is updated using a four site rule with two control bits and two action bits. The model displays rich dynamics. There are three types of stable particles, left movers, right movers and ‘frozen’ bound states that only move as an effect of scattering with the left and right movers. Multi-particle scattering in the system is factorized. We embed the model into the canonical framework of Yang–Baxter integrability by rigorously proving the existence of a commuting set of diagonal-to-diagonal transfer matrices. The construction involves a new type of Lax operator.
We wrap up the main outcomes of our investigations. Even if not all questions and issues could be addressed as precisely and persuasively as one would wish, we do find that the assumption of an underlying theory describing a single, definite world with completely deterministic evolution laws, can be maintained and pursued further. In the more distant future, it is hoped that the Cellular Automaton Theory will enable us to select credible models for grand unification an quantized gravity; in the mean while, the Cellular Automaton Interpretation will be helpful in visualizing what we are actually doing when we are involved with quantum mechanical calculations. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
Even if we still face problems in formulating concise theories that explain the resolution of various apparent paradoxes, one may simply assume such a theory to exist, and that it will one day be found. This leads to the CA (Cellular Automaton) Interpretation. We now formulate the rules. The universe has a wave function, which, in terms of the classical degrees of freedom, collapses continuously, yet fully in agreement with Schrödinger’s equation. We formulate how to look at various notoriously ‘quantum mechanical’ phenomena, such as superposition, exponential decay, and the double slit experiment. What can we say about the ‘quantum computer’? These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
The Thirring Quantum Cellular Automaton (QCA) describes the discrete time dynamics of local fermionic modes that evolve according to one step of the Dirac cellular automaton, followed by the most general on-site number-preserving interaction, and serves as the QCA counterpart of the Thirring model in quantum field theory. In this work, we develop perturbative techniques for the QCA path sum approach, expanding both the number of interaction vertices and the mass parameter of the Thirring QCA. By classifying paths within the regimes of very light and very heavy particles, we computed the transition amplitudes in the two- and three-particle sectors to the first few orders. Our investigation into the properties of the Thirring QCA, addressing the combinatorial complexity of the problem, yielded some useful results applicable to the many-particle sector of any on-site number-preserving interactions in one spatial dimension.
Simulations of one quantum system by an other has an implication in realization of quantum machine that can imitate any quantum system and solve problems that are not accessible to classical computers. One of the approach to engineer quantum simulations is to discretize the space-time degree of freedom in quantum dynamics and define the quantum cellular automata (QCA), a local unitary update rule on a lattice. Different models of QCA are constructed using set of conditions which are not unique and are not always in implementable configuration on any other system. Dirac Cellular Automata (DCA) is one such model constructed for Dirac Hamiltonian (DH) in free quantum field theory. Here, starting from a split-step discrete-time quantum walk (QW) which is uniquely defined for experimental implementation, we recover the DCA along with all the fine oscillations in position space and bridge the missing connection between DH-DCA-QW. We will present the contribution of the parameters resulting in the fine oscillations on the Zitterbewegung frequency and entanglement. The tuneability of the evolution parameters demonstrated in experimental implementation of QW will establish it as an efficient tool to design quantum simulator and approach quantum field theory from principles of quantum information theory.
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.