Projecting three-dimensional color codes onto three-dimensional toric codes

Toric codes and color codes are two important classes of topological codes. Kubica et al. [A. Kubica, New J. Phys. 17, 083026 (2015)NJOPFM1367-263010.1088/1367-2630/17/8/083026] showed that any D-dimensional color code can be mapped to a finite number of toric codes in D dimensions. We propose an alternate map of three-dimensional (3D) color codes to 3D toric codes with a view to decoding 3D color codes. Our approach builds on Delfosse's result [N. Delfosse, Phys. Rev. A 89, 012317 (2014)PLRAAN1050-294710.1103/PhysRevA.89.012317] 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. We use these projections to study the performance of a class of 3D color codes called stacked codes.