# Anomalous Subdiffusion from Subsystem Symmetries

## TL;DR

This paper introduces automaton quantum circuits with subsystem symmetries that exhibit anomalous subdiffusion, revealing slower-than-diffusive charge spreading and non-Gaussian distributions, supported by numerical simulations and operator hydrodynamics.

## Contribution

It characterizes automaton quantum circuits with subsystem symmetries and demonstrates their role in causing anomalous subdiffusion and non-Gaussian charge distributions in quantum dynamics.

## Key findings

- Charge autocorrelator decays as log(t)/√t in 2D with line symmetries
- Charge autocorrelator decays as 1/t^{3/4} in 3D with planar symmetries
- Numerical evidence of non-Gaussian charge distributions and quantum chaos signatures

## Abstract

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.