  Quantum cellular automata (QCA) evolve qubits in a quantum circuit depending
only on the states of their neighborhoods and model how rich physical
complexity can emerge from a simple set of underlying dynamical rules. For
instance, Goldilocks QCA depending on trade-off principles exhibit
non-equilibrating coherent dynamics and generate complex mutual information
networks, much like the brain. The inability of classical computers to simulate
large quantum systems is a hindrance to understanding the physics of quantum
cellular automata, but quantum computers offer an ideal simulation platform.
Here we demonstrate the first experimental realization of QCA on a digital
quantum processor, simulating a one-dimensional Goldilocks rule on chains of up
to 23 superconducting qubits. Employing low-overhead calibration and error
mitigation techniques, we calculate population dynamics and complex network
measures indicating the formation of small-world mutual information networks.
Unlike random states, these networks decohere at fixed circuit depth
independent of system size; the largest of which corresponds to 1,056 two-qubit
gates. Such computations may open the door to the employment of QCA in
applications like the simulation of strongly-correlated matter or
beyond-classical computational demonstrations.
