Projecting 3D color codes onto 3D 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.

arXiv commentLaTeX, 11 pages. Improved presentation, simplified proofs, main results unchanged

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• Anyon condensation and the color code
• Parallelized quantum error correction with fracton topological codes
• A decoder for the triangular color code by matching on a Möbius strip
• Efficient color code decoders in $d\ge 2$ dimensions from toric code decoders
• Quantum Pin Codes
• Efficiently decoding the 3D toric codes and welded codes on cubic lattices
• Layer-by-layer disentangling two-dimensional topological quantum codes
• Pseudocodeword-based Decoding of Quantum Color Codes