Decoder for the Triangular Color Code by Matching on a Möbius Strip

Abstract

The color code is remarkable for its ability to perform fault-tolerant logic gates. This motivates the design of practical decoders that minimize the resource cost of color-code quantum computation. Here we propose a decoder for the planar color code with a triangular boundary where we match syndrome defects on a nontrivial manifold that has the topology of a Möbius strip. A basic implementation of our decoder used on the color code with hexagonal lattice geometry demonstrates a logical failure rate that is competitive with the optimal performance of the surface code. The logical failure rate scales approximately like p^{αsqrt[n]}, with α≈6/7sqrt[3]≈0.5, error rate p, and n the code length. Furthermore, by exhaustively testing over five billion error configurations, we find that a modification of our decoder that manually compares inequivalent recovery operators can correct all errors of weight ≤(d−1)/2 for codes with distance d≤13. Our decoder is derived using relations among the stabilizers that preserve global conservation laws at the lattice boundary. We present generalizations of our method to depolarizing noise and fault-tolerant error correction, as well as to Majorana surface codes, higher-dimensional color codes, and single-shot error correction.