  We introduce quantum circuits in two and three spatial dimensions which are
classically simulable, despite producing a high degree of operator
entanglement. We provide a partial characterization of these "automaton"
quantum circuits, and use them to study operator growth, information spreading,
and local charge relaxation in quantum dynamics with subsystem symmetries,
which we define as overlapping symmetries that act on lower-dimensional
submanifolds. With these symmetries, we discover the anomalous subdiffusion of
conserved charges; that is, the charges spread slower than diffusion in the
dimension of the subsystem symmetry. By studying an effective operator
hydrodynamics in the presence of these symmetries, we predict the charge
autocorrelator to decay ($i$) as $\log(t)/\sqrt{t}$ in two dimensions with a
conserved $U(1)$ charge along intersecting \emph{lines}, and ($ii$) as
$1/t^{3/4}$ in three spatial dimensions with intersecting \emph{planar} $U(1)$
symmetries. Through large-scale studies of automaton dynamics with these
symmetries, we numerically observe charge relaxation that is consistent with
these predictions. In both cases, the spatial charge distribution is distinctly
non-Gaussian, and reminiscent of the diffusion of charges along a fractal
surface. We numerically study the onset of quantum chaos in the spreading of
local operators under these automaton dynamics, and observe power-law
broadening of the ballistically-propagating fronts of evolving operators in two
and three dimensions, and the saturation of out-of-time-ordered correlations to
values consistent with quantum chaotic behavior.
