Decoding the three-dimensional toric codes and welded codes on cubic lattices

Recent years have seen a growing interest in quantum codes in three dimensions (3D). One of the earliest proposed 3D quantum codes is the 3D toric code. It has been shown that 3D color codes can be mapped to 3D toric codes. The 3D toric code on a cubic lattice is also a building block for the welded code which has the highest energy barrier to date. Although well known, the performance of the 3D toric code has not been studied extensively. In this paper we propose efficient decoding algorithms for the 3D toric code on a cubic lattice with and without boundaries and report their performance for various quantum channels. We observe a threshold of -12% for the bit flip errors, ≈3% for the phase flip errors, and 24.8% for the erasure channel. We also study the performance of the welded 3D toric code on the quantum erasure channel. We did not observe a threshold for the welded code over the erasure channel for the parameters we simulated.