Fault-tolerant renormalization group decoder for abelian topological codes

We present a three-dimensional generalization of a renormalization group decoding algorithm for topological codes with Abelian anyonic excitations that we introduced for two dimensions in DP09a, DP10a. We also provide a complete detailed description of the structure of the algorithm, which should be sufficient for anyone interested in implementing it. This 3D implementation extends our previous 2D algorithm by incorporating a failure probability of the syndrome measurements, i. e. , it enables fault-tolerant decoding. We report a fault-tolerant storage threshold of 1. 9 (4) \% for Kitaev's toric code subject to a 3D bit-flip channel (i. e. including imperfect syndrome measurements). This number is to be compared with the 2. 9\% value obtained via perfect matching H04a. The 3D generalization inherits many properties of the 2D algorithm, including a complexity linear in the space-time volume of the memory, which can be parallelized to logarithmic time.