Critical Quantum Dynamics in $\mathbf{(1+1)}$-dimensional Quantum Cellular Automata with Projected Entangled Pair States.

Motivated by recent progress in the experimental development of quantum simulators based on Rydberg atoms, we investigate the dynamics of a class of $1+1$-dimensional quantum cellular automata. Their key feature is that they display stationary behavior and non-equilibrium phase transitions despite being isolated systems. We show that a tensor network representation through projected entangled pair states permits an efficient encoding of the cellular automata dynamics. This representation, which naturally captures the structure of the considered cellular automata, also reflects the degree of quantumness and complexity of the dynamics in the difficulty of contracting the tensor network. To illustrate this, we explore the role of quantum correlations in the critical physics emerging in a class of cellular automata which offers a controllable degree of local entanglement together with a steady-state non-equilibrium phase transition. Due to its flexibility, the framework we introduce is promising, both for applications to the study of quantum non-equilibrium phase transitions and for investigation of quantum correlations in complex dynamics.