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Physical Review Letters

Quantum and Classical Temporal Correlations in (1+1)D Quantum Cellular Automata

Edward Gillman, Federico Carollo, Igor Lesanovsky

Published Dec 1, 2021• 14 citations
Publisher

Abstract

We employ ($1+1$)-dimensional quantum cellular automata to study the evolution of entanglement and coherence near criticality in quantum systems that display nonequilibrium steady-state phase transitions. This construction permits direct access to the entire space-time structure of the underlying nonequilibrium dynamics, and allows for the analysis of unconventional correlations, such as entanglement in the time direction between the ``present'' and the ``past.'' We show how the uniquely quantum part of these correlations---the coherence---can be isolated and that, close to criticality, its dynamics displays a universal power-law behavior on approach to stationarity. Focusing on quantum generalizations of classical nonequilibrium systems: the Domany-Kinzel cellular automaton and the Bagnoli-Boccara-Rechtman model, we estimate the universal critical exponents for both the entanglement and coherence. As these models belong to the one-dimensional directed percolation universality class, the latter provides a key new critical exponent, one that is unique to quantum systems.

DOI
10.1103/physrevlett.127.230502

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We employ (1+11+11+1)-dimensional quantum cellular automata to study the evolution of entanglement and coherence near criticality in quantum systems that display nonequilibrium steady-state phase transitions. This construction permits direct access to the entire space-time structure of the underlying nonequilibrium dynamics, and allows for the analysis of unconventional correlations, such as entanglement in the time direction between the present'' and the past.'' We show how the uniquely quantum part of these correlations---the coherence---can be isolated and that, close to criticality, its dynamics displays a universal power-law behavior on approach to stationarity. Focusing on quantum generalizations of classical nonequilibrium systems: the Domany-Kinzel cellular automaton and the Bagnoli-Boccara-Rechtman model, we estimate the universal critical exponents for both the entanglement and coherence. As these models belong to the one-dimensional directed percolation universality class, the latter provides a key new critical exponent, one that is unique to quantum systems.
Quantum and classical temporal correlations in (1+1)D(1 + 1)D(1+1)D Quantum Cellular Automata
We employ (1+1)-dimensional quantum cellular automata to study the evolution of entanglement and coherence near criticality in quantum systems that display nonequilibrium steady-state phase transitions. This construction permits direct access to the entire space-time structure of the underlying nonequilibrium dynamics, and allows for the analysis of unconventional correlations, such as entanglement in the time direction between the "present" and the "past." We show how the uniquely quantum part of these correlations-the coherence-can be isolated and that, close to criticality, its dynamics displays a universal power-law behavior on approach to stationarity. Focusing on quantum generalizations of classical nonequilibrium systems: the Domany-Kinzel cellular automaton and the Bagnoli-Boccara-Rechtman model, we estimate the universal critical exponents for both the entanglement and coherence. As these models belong to the one-dimensional directed percolation universality class, the latter provides a key new critical exponent, one that is unique to quantum systems.
Classical (1+1)(1+1)(1+1)-dimensional (D) cellular automata, as for instance Domany-Kinzel cellular automata, are paradigmatic systems for the study of nonequilibrium phenomena. Such systems evolve in discrete time steps, and are thus free of time-discretization errors. Moreover, they display nonequilibrium phase transitions which can be studied by simulating the evolution of an initial seed. At any finite time, this has support only on a finite light cone. Thus, essentially numerically exact simulations free of finite-size errors or boundary effects are possible, leading to high-accuracy estimates of critical exponents. Here, we show how similar advantages can be gained in the quantum regime: The many-body critical dynamics occurring in (1+1)D(1+1)\mathrm{D}(1+1)D quantum cellular automata with an absorbing state can be studied directly on an infinite lattice when starting from seed initial conditions. This can be achieved efficiently by simulating the dynamics of an associated one-dimensional, nonunitary quantum cellular automaton using tensor networks. We apply our method to a model introduced recently and find accurate values for universal exponents, suggesting that this approach can be a powerful tool for precisely studying nonequilibrium universal physics in quantum systems.
Motivated by recent progress in the experimental development of quantum simulators based on Rydberg atoms, we introduce and investigate the dynamics of a class of (1+1)-dimensional quantum cellular automata. These nonequilibrium many-body models, which are quantum generalizations of the Domany-Kinzel cellular automaton, possess two key features: they display stationary behavior and nonequilibrium phase transitions despite being isolated systems. Moreover, they permit the controlled introduction of local quantum correlations, which allows for the impact of quantumness on the dynamics and phase transition to be assessed. We show that projected entangled pair state tensor networks permit a natural and efficient representation of the cellular automaton. Here, the degree of quantumness and complexity of the dynamics is reflected in the difficulty of contracting the tensor network.
Probabilistic cellular automata (CA) provide a classic framework for studying nonequilibrium statistical physics on lattices. A notable example is the Domany-Kinzel CA, which has been used to investigate the process of directed percolation and the critical dynamics of the nonequilibrium phase transition between absorbing and percolating phases. In this work, we construct a nonunitary quantum CA that generalizes the Domany-Kinzel CA and study the resulting dynamical evolution using numerical simulations using the tensor network infinite time-evolving block decimation (iTEBD) algorithm. We demonstrate that the system undergoes the absorbing/percolating phase transition and that the addition of the Hamiltonian generates coherences, which are a distinct feature of quantum dynamics. A proposal for the implementation of the model with Rydberg array is put forward, which does not require local addressing of individual sites.
Probabilistic cellular automata provide a simple framework for exploring classical nonequilibrium processes. Recently, quantum cellular automata have been proposed that rely on the propagation of a one-dimensional quantum state along a fictitious discrete time dimension via the sequential application of quantum gates. The resulting (1+1)-dimensional space-time structure makes these automata special cases of recurrent quantum neural networks which can implement broad classes of classical nonequilibrium processes. Here, we present a general prescription by which these models can be extended into genuinely quantum nonequilibrium models via the systematic inclusion of asynchronism. This is illustrated for the classical contact process, where the resulting model is closely linked to the quantum contact process (QCP), developed in the framework of open quantum systems. Studying the mean-field behavior of the model, we find evidence of an "asynchronism transition," i.e., a sudden qualitative change in the phase transition behavior once a certain degree of asynchronicity is surpassed, a phenomenon we link to observations in the QCP.
Quantum devices featuring mid-circuit measurement and reset capabilities, such as quantum computers and dual-species Rydberg quantum simulators, enable the realization of quantum cellular automata. These systems evolve in discrete time following local updates implemented by unitary gates, and allow for the realization of both closed and synthetic dissipative dynamics. Here, we focus on quantum cellular automata that implement minimal models of classical and quantum transport. To illustrate our ideas, we focus on a discrete-time totally asymmetric simple exclusion process and investigate how coherent dynamical contributions allow for the emergence of quantum effects and correlations. We find that bipartite entanglement dominates the transient evolution, while stationary states can retain quantum correlations beyond entanglement. Our results suggest viable routes for realizing transport models on quantum devices and characterizing collective quantum correlations in strongly driven systems.