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We consider the dynamics of local entropy and nearest neighbor mutualinformation of a 1-D lattice of qubits via the repeated application of nearestneighbor CNOT quantum gates. This is a quantum version of a cellular automaton.We analyze the entropy dynamics for different initial product states, both foropen boundary conditions, periodic boundary conditions and we also consider theinfinite lattice thermodynamic limit. The dynamics gives rise to fractalbehavior, where we see the appearance of the Sierpinski triangle both forstates in the computational basis and for operator dynamics in the Heisenbergpicture. In the thermodynamics limit, we see equilibration with a timedependence controlled by $\exp(-\alpha t^{h-1})$ where $h$ is the fractaldimension of the Sierpinski triangle, and $\alpha$ depends on the details ofthe initial state. We also see log-periodic reductions in the one qubit entropywhere the approach to equilibrium is only power law. For open boundaryconditions we see time periodic oscillations near the boundary, associated tosubalgebras of operators localized near the boundary that are mapped tothemselves by the dynamics.

Exotic equilibration dynamics on a 1-D quantum CNOT gate lattice