Decoding color codes by projection onto surface codes

TL;DR

This paper introduces a novel decoding strategy for color codes by projecting errors onto surface codes, enabling the use of existing algorithms and establishing error threshold bounds.

Contribution

It presents a new projection-based decoding method for color codes, leveraging surface code algorithms and chain complex theory, with practical threshold estimates.

Findings

Achieved an error threshold of approximately 8.7% for hexagonal color codes.

Provided a general lower bound on color code error thresholds based on surface code thresholds.

Demonstrated the applicability of surface code decoders to color codes through projection.

Abstract

We propose a new strategy to decode color codes, which is based on the projection of the error onto three surface codes. This provides a method to transform every decoding algorithm of surface codes into a decoding algorithm of color codes. Applying this idea to a family of hexagonal color codes, with the perfect matching decoding algorithm for the three corresponding surface codes, we find a phase error threshold of approximately 8.7%. Finally, our approach enables us to establish a general lower bound on the error threshold of a family of color codes depending on the threshold of the three corresponding surface codes. These results are based on a chain complex interpretation of surface codes and color codes.