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The recent years have seen a growing interest in quantum codes in threedimensions (3D). One of the earliest proposed 3D quantum codes is the 3D toriccode. 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 codewhich has highest energy barrier to date. Although well known, the performanceof the 3D toric code has not been studied extensively. In this paper, wepropose efficient decoding algorithms for the 3D toric code on a cubic latticewith and without boundaries and report their performance for various quantumchannels. 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 alsostudy the performance of the welded 3D toric code on the quantum erasurechannel. We did not observe a threshold for the welded code over the erasurechannel.

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