Cellular-automaton decoders with provable thresholds for topological codes

TL;DR

This paper introduces a new cellular automaton-based decoder, the Sweep Decoder, for topological quantum codes, providing provable error correction thresholds and demonstrating its effectiveness through numerical estimates.

Contribution

The paper develops the Sweep Rule cellular automaton and the Sweep Decoder, extending Toom's rule to higher dimensions and proving its threshold performance for topological codes.

Findings

The Sweep Decoder has a provable lower bound on performance.

Numerical estimates show the Sweep Decoder threshold for 3D toric codes.

The approach applies to various topological codes like the color code.

Abstract

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.