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arXiv:2105.01394 · Physical Review Research

Directed percolation in nonunitary quantum cellular automata

Ramil Nigmatullin, Elisabeth Wagner, Gavin K. Brennen

Published Dec 10, 2021• 16 citations
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Abstract

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.

DOI
10.1103/physrevresearch.3.043167

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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.
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.
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.
Changes in the entanglement structure and critical phenomena are hallmarks of quantum phase transitions. Here, we discuss how they appear in transitions between classes of states with distinct entanglement patterns beyond the paradigm of stable equilibrium phases of matter. Using a mapping between stochastic automata and isometric Tensor Network States (isoTNS), we construct a two-dimensional quantum state from the Domany-Kinzel automaton, which is a (1+1)D process with an absorbing phase transition in the directed percolation class. At the critical point of the automaton, the corresponding isoTNS hosts algebraic correlations in all spatial directions. The continuous parent Hamiltonian of this state has a degenerate ground state manifold. It consists of a product state (the absorbing state) and a second state that undergoes a transition from pairwise entanglement between distant regions, similarly to the W state, to a state with trivial entanglement. Our results demonstrate how the correspondence between isoTNS and classical stochastic evolution can be used to probe the Hilbert space structure beyond stable ground state manifolds.
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.
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.
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.