Exact Thermalization Dynamics in the “Rule 54” Quantum Cellular Automaton

We study the out-of-equilibrium dynamics of the quantum cellular automaton known as “Rule 54.” For a class of low-entangled initial states, we provide an analytic description of the effect of the global evolution on finite subsystems in terms of simple quantum channels, which gives access to the full thermalization dynamics at the microscopic level. As an example, we provide analytic formulas for the evolution of local observables and Rényi entropies. We show that, in contrast to other known examples of exactly solvable quantum circuits, Rule 54 does not behave as a simple Markovian bath on its own parts, and displays typical nonequilibrium features of interacting integrable many-body quantum systems such as finite relaxation rate and interaction-induced dressing effects. Our study provides a rare example where the full thermalization dynamics can be solved exactly at the microscopic level.

Figure 1

Quantum circuit representation of the Rule 54 quantum cellular automaton. The dynamics can be equivalently given in terms of three-site gates (left) and as an MPO (right).

Figure 2

Operator evolution in the MPO form starting from an MPS and the folding procedure. Time evolution of operators can be efficiently represented by combining the tensors and their complex conjugate into a supertensor acting on the doubled space.

Figure 3

Growth of Rényi entropies, for $n=2$, 3, 4 and $\phi =0$. The values of $S_n(t)$ are rescaled with time.