  We present a three-dimensional generalization of a renormalization group
decoding algorithm for topological codes with Abelian anyonic excitations that
we previously introduced for two dimensions. 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. 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.
