Cellular-automaton decoders with provable thresholds for topological codes

Understand

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 $d\geq 3$ 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.