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Our algorithm has a worst case complexity of $O(n \\alpha(n))$, where $n$ is the number of physical qubits and $\\alpha$ is the inverse of Ackermann's function, which is very slowly growing. For all practical purposes, $\\alpha(n) \\leq 3$. We prove that our algorithm performs optimally for errors of weight up to $(d-1)/2$ and for loss of up to $d-1$ qubits, where $d$ is the minimum distance of the code. Numerically, we obtain a threshold of $9.9\\%$ for the 2d-toric code with perfect syndrome measurements and $2.6\\%$ with faulty measurements.","url_abs":"https://arxiv.org/abs/1709.06218v3","url_pdf":"https://arxiv.org/pdf/1709.06218v3.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":"almost-linear-time-decoding-algorithm-for","repo_url":"https://github.com/chaeyeunpark/UnionFind","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":null},{"paper_slug":"almost-linear-time-decoding-algorithm-for","repo_url":"https://github.com/poeloe/oop_surface_code","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":{"status":"unanswered"}},{"paper_slug":"almost-linear-time-decoding-algorithm-for","repo_url":"https://github.com/siddhantphy/qsurface","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":{"status":"unanswered"}},{"paper_slug":"almost-linear-time-decoding-algorithm-for","repo_url":"https://github.com/watermarkhu/qsurface","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":{"status":"unanswered"}}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":"https://app.syntology.ai/?focus=1709.06218","mcp":{"get_harvested_code_for_paper":{"arxiv_id":"1709.06218"}},"developers":"https://syntology.ai/developers","read_at":"2026-09-24T18:15:14+00:00","read_at_is":"when the build read Syntology's graph, not when any sample ran","claim":"Per-sample execution status on synthesized fixtures; not a correctness claim about the paper. 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