Projecting 3D color codes onto 3D toric codes

TL;DR

This paper introduces a new method to map 3D color codes onto 3D toric codes, simplifying the decoding process by leveraging topological properties and building on previous 2D results.

Contribution

It presents an alternative mapping of 3D color codes to 3D toric codes aimed at improving decoding strategies for 3D color codes.

Findings

Decoding 3D color codes reduces to decoding 3D toric codes.

Bit flip errors are decoded via projection onto one set of toric codes.

Phase flip errors are decoded via projection onto another set of toric codes.

Abstract

Toric codes and color codes are two important classes of topological codes. Kubica, Yoshida, and Pastawski showed that any $D$-dimensional color code can be mapped to a finite number of toric codes in $D$-dimensions. In this paper we propose an alternate map of 3D color codes to 3D toric codes with a view to decoding 3D color codes. Our approach builds on Delfosse's result for 2D color codes and exploits the topological properties of these codes. Our result reduces the decoding of 3D color codes to that of 3D toric codes. Bit flip errors are decoded by projecting on one set of 3D toric codes while phase flip errors are decoded by projecting onto another set of 3D toric codes.