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arXiv:2006.02440 · Physical review. B./Physical review. B

Nonergodic quantum dynamics from deformations of classical cellular automata

Thomas Iadecola, Sagar Vijay

Published Nov 6, 2020• 43 citations
arXivHTMLPDF

Abstract

Classical reversible cellular automata (CAs), which describe the discrete-time dynamics of classical degrees of freedom in a finite state space, can exhibit exact, nonthermal quantum eigenstates despite being classically chaotic. We show that every classical CA defines a family of generically nonintegrable, periodically driven (Floquet) quantum dynamics with exact, nonthermal eigenstates. These Floquet dynamics are nonergodic in the sense that certain product states on a periodic classical orbit fail to thermalize, while generic initial states thermalize as expected in a quantum chaotic system. We demonstrate that some signatures of these effects can be probed in quantum simulators based on Rydberg atoms in the blockade regime. These results establish classical CAs as parent models for a class of quantum chaotic systems with rare nonthermal eigenstates.

DOI
10.1103/physrevb.102.180302

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Classical reversible cellular automata (CAs), which describe the discrete-time dynamics of classical degrees of freedom in a finite state space, can exhibit exact, nonthermal quantum eigenstates despite being classically chaotic. We show that every classical CA defines a family of generically nonintegrable, periodically driven (Floquet) quantum dynamics with exact, nonthermal eigenstates. These Floquet dynamics are nonergodic in the sense that certain product states on a periodic classical orbit fail to thermalize, while generic initial states thermalize as expected in a quantum chaotic system. We demonstrate that some signatures of these effects can be probed in quantum simulators based on Rydberg atoms in the blockade regime. These results establish classical CAs as parent models for a class of quantum chaotic systems with rare nonthermal eigenstates.
We study the ergodic properties of a unitary Floquet dynamics arising from the repeated application of a translationally-invariant Clifford Quantum Cellular Automata to an infinite system of qubits in d dimensions. One expects that if the QCA does not exhibit any periodicity, a generic initial state of qubits will thermalize, that is, approach the infinite-temperature state. We show that this is true for many classes of states, both pure and mixed. In particular, this is true for all initial states that are short-range entangled and close to the equilibrium state. We also point out a subtle distinction between weak and strong thermalization.
Abstract We introduce and describe a class of simple facilitated quantum spin models in which the dynamics is due to the repeated application of unitary gates. The gates are applied periodically in time, so their combined action constitutes a Floquet unitary. The dynamics of the models we discuss can be classically simulated, and their eigenstates classically constructed (despite being highly entangled). We consider a variety of models in one and two dimensions: we are primarily concerned with models involving Clifford gates, but we also briefly discuss a model involving Clifford and Toffoli gates. Two of these models are integrable free-particle models; we construct their conserved densities. For all five models, however, the eigenstate entanglement (in all cases) and operator dynamics (in all but one case) differ from those of generic chaotic systems. In addition, two of these models have exponentially many eigenstates in which one or more sites ‘disentangle’ from the rest of the system, as a consequence of reflection symmetry.
Ergodicity breaking is observed in the blockade regime of Rydberg atom arrays, in the form of low entanglement eigenstates known as scars, which fail to thermalize. The signature of these states persists in periodically driven systems, where they coexist with an extensive number of chaotic states. Here we investigate a quantum cellular automaton based on the classical rule that updates a site if its two neighbors are in the lower state. We show that the breaking of ergodicity extends to chaotic states. The dynamical breaking of ergodicity is controlled by chiral quasiparticle excitations which propagate entanglement. Evidence of nonlocal entanglement is found, showing that these nonthermal chaotic states may be useful to quantum computation.
We introduce a method of quantum emulation of a classical reversible cellular automaton. By applying this method to a chaotic cellular automaton, the obtained quantum many-body system thermalizes while all the energy eigenstates and eigenvalues are solvable. These explicit solutions allow us to verify the validity of some scenarios of thermalization to this system. We find that two leading scenarios, the eigenstate thermalization hypothesis scenario and the large effective dimension scenario, do not explain thermalization in this model.
This paper investigates the potential for emergent computation within cellular automata (CA) systems incorporating quantum mechanical principles. Traditional CAs, while capable of simulating complex patterns, are fundamentally deterministic. We propose an extension of the CA framework where individual cells possess quantum properties, specifically superposition and entanglement, influencing their interactions. This introduces a layer of probabilistic behavior and non-classical correlations, fundamentally altering the system's dynamics. The core claim is that these quantum interactions can lead to the emergence of computational processes previously unattainable within purely classical CAs. We detail a model where cells exist in a superposition of states, and interactions are mediated through entangled states, allowing for correlations that defy classical explanation. We explore the theoretical implications of such a system, outlining potential computational capabilities and challenges associated with simulating quantum effects within the discrete framework of a CA. The work highlights a novel approach to computation, suggesting a pathway toward harnessing quantum phenomena for emergent complex behavior in simulated systems.
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.