Efficiently decoding the 3D toric codes and welded codes on cubic lattices

Abstract: The 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 cubic lattice is also a building block for the welded code which has 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 $\gtrsim 12\%$ for the bit flip errors, $\approx 3\%$ for phase flip errors and $ 24.8\%$ for 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.