A 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 minimise 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, $\sim p^{\alpha \sqrt{n}}$, with $\alpha \approx 6/7\sqrt{3}\approx 0.5$, 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 $\le (d-1)/2$ for codes with distance $d\le 13$. Our decoder is derived using relations among the stabilizers that preserve global conservation laws at the lattice boundary. We present generalisations of our method to depolarising noise and fault-tolerant error correction, as well as to Majorana surface codes, higher-dimensional color codes and single-shot error correction.

31 pages, 27 figures, comments welcome; v2 - references added, typos corrected, revised discussion about exhaustive search, conclusions remain the same; v3 - final author version, changes made in response to peer review to improve clarity, conclusions unchanged

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Cited by in corpus (4)

• Anyon condensation and the color code
• A local pre-decoder to reduce the bandwidth and latency of quantum error correction
• Conservation laws and quantum error correction: towards a generalised matching decoder
• Low-overhead quantum computing with the color code