{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/a-decoder-for-the-triangular-color-code-by","title":"A decoder for the triangular color code by matching on a Möbius strip","arxiv_id":"2108.11395","date":"2021-08-25","proceeding":null,"authors":["Kaavya Sahay","Benjamin J. Brown"],"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 M\\\"{o}bius 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.","url_abs":"https://arxiv.org/abs/2108.11395v3","url_pdf":"https://arxiv.org/pdf/2108.11395v3.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"a-decoder-for-the-triangular-color-code-by","repo_url":"https://github.com/quantumlib/chromobius","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}