Cellular-Automaton Decoders with Provable Thresholds for Topological Codes.

We propose a new cellular automaton (CA), the sweep rule, which generalizes Toom's rule to any locally Euclidean lattice. We use the sweep rule to design a local decoder for the toric code in d≥3 dimensions, the sweep decoder, and rigorously establish a lower bound on its performance. We also numerically estimate the sweep decoder threshold for the three-dimensional toric code on the cubic and body-centered cubic lattices for phenomenological phase-flip noise. Our results lead to new CA decoders with provable error-correction thresholds for other topological quantum codes including the color code.