Nonequilibrium Phase Transitions in ($1+1$)-Dimensional Quantum Cellular Automata with Controllable Quantum Correlations

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.

Figure 1

The $(1+1)D$ QCA. (a) The dynamics of the $(1+1)D$ QCA is enacted by sweeping through row by row with the a fundamental three-body gate $G$. This generates an effective time dimension in the vertical direction. The initial state is chosen to be empty at all sites apart from the first row. (b) $G$ is controlled by two sites, labeled by 1 and 2, and acts on a target one, denoted by 3. This controlled operation defines the elementary propagation block. (c) Single-row observables are fully characterized by the reduced state of a row $\rho (t)$ at the discrete time $t$. (d) When $G$ has an invariant configuration, the $(1+1)D$ QCA features an absorbing state.

Figure 2

PEPS representation for $(1+1)D$ QCA: (a) The three-body gates $G$ can be written as a three-site matrix product operator (MPO) with maximum bond dimension 4. (b) The initial state is represented by an unentangled PEPS. As the MPO representation of $G$ is applied, at time step $t$ of the evolution, each bond in the PEPS has at most bond dimension 4 for rows up to $t+1$, and bond dimension 1 otherwise. (c) Expectation values are represented by the double-layer TN (DLTN) obtained by sandwiching the chosen operator between the PEPS representation of the $2D$ state and its dual. (d) To evaluate the expectation, the DLTN must be contracted; this operation cannot be done exactly in general and requires approximation.

Figure 3

Gate entangling power and dynamics of the QCA.—(a) The entangling effect of the gate is quantified via the concurrence of the reduced density matrix of the two target sites following the action of two gates on three control sites in the state $|\circ •\circ ⟩$. (b) Density plot of the concurrence of the reduced density matrix of the target sites. For $\omega =0$, $\pi /2$ there is no entanglement, whatever the value of $\mathrm{\Gamma }$. (c) Average density at the $t$th discrete time. The approximation is obtained using the boundary MPS method with largest bond dimension $\chi =256$ or 512 and up to $t=100$. We explore the range $\omega \in [0,\pi /2]$ with values of $\mathrm{\Gamma }$ including both subcritical and supercritical curves—identified by negative and positive curvatures in the log-log plot, respectively. This reveals the NEPT of the QCA. When $\omega =0$, 1.55, all curves for different $\chi$ values are indistinguishable on the scale shown. This contrasts the case of intermediate $\omega$ where larger $\chi$ values are required for convergence. This is broadly in line with the emergence of nonzero concurrence in panel (b).

Figure 4

Phase diagram of the $(1+1)D$ QCA. The stationary state density $n_{\mathrm{ss}}$ estimated using a (mean-field) MF approach, is shown as a contour-plot with the indicated colour map. The corresponding MF phase boundary is displayed as a red solid line, and shows no dependence on $\omega$. By contrast, the phase boundary extracted from PEPS simulations (yellow circles) shows that the critical $\mathrm{\Gamma }$ does indeed depend on the parameter $\omega$ that controls the local entanglement.