Local Decoders for the 2D and 4D Toric Code

We analyze the performance of decoders for the 2D and 4D toric code which are local by construction. The 2D decoder is a cellular automaton decoder formulated by Harrington which explicitly has a finite speed of communication and computation. For a model of independent X and Z errors and faulty syndrome measurements with identical probability we report a threshold of 0.133% for this decoder. We implement a decoder for the 4D toric code which is based on a decoder by Hastings.