  We consider the dynamics of local entropy and nearest neighbor mutual
information of a 1-D lattice of qubits via the repeated application of nearest
neighbor CNOT quantum gates. This is a quantum version of a cellular automaton.
We analyze the entropy dynamics for different initial product states, both for
open boundary conditions, periodic boundary conditions and we also consider the
infinite lattice thermodynamic limit. The dynamics gives rise to fractal
behavior, where we see the appearance of the Sierpinski triangle both for
states in the computational basis and for operator dynamics in the Heisenberg
picture. In the thermodynamics limit, we see equilibration with a time
dependence controlled by $\exp(-\alpha t^{h-1})$ where $h$ is the fractal
dimension of the Sierpinski triangle, and $\alpha$ depends on the details of
the initial state. We also see log-periodic reductions in the one qubit entropy
where the approach to equilibrium is only power law. For open boundary
conditions we see time periodic oscillations near the boundary, associated to
subalgebras of operators localized near the boundary that are mapped to
themselves by the dynamics.
