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Our algorithm has a worst case complexity of <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mi>O<\/mml:mi><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>n<\/mml:mi><mml:mi>\u03b1<\/mml:mi><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>n<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:math>, where <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mi>n<\/mml:mi><\/mml:math> is the number of physical qubits and <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mi>\u03b1<\/mml:mi><\/mml:math> is the inverse of Ackermann's function, which is very slowly growing. For all practical purposes, <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mi>\u03b1<\/mml:mi><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>n<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><mml:mo>\u2264<\/mml:mo><mml:mn>3<\/mml:mn><\/mml:math>. We prove that our algorithm performs optimally for errors of weight up to <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>d<\/mml:mi><mml:mo>\u2212<\/mml:mo><mml:mn>1<\/mml:mn><mml:mo stretchy=\"false\">)<\/mml:mo><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo>\/<\/mml:mo><\/mml:mrow><mml:mn>2<\/mml:mn><\/mml:math> and for loss of up to <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mi>d<\/mml:mi><mml:mo>\u2212<\/mml:mo><mml:mn>1<\/mml:mn><\/mml:math> qubits, where <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mi>d<\/mml:mi><\/mml:math> is the minimum distance of the code. Numerically, we obtain a threshold of <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mn>9.9<\/mml:mn><mml:mi mathvariant=\"normal\">%<\/mml:mi><\/mml:math> for the 2d-toric code with perfect syndrome measurements and <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mn>2.6<\/mml:mn><mml:mi mathvariant=\"normal\">%<\/mml:mi><\/mml:math> with faulty measurements.<\/jats:p>","DOI":"10.22331\/q-2021-12-02-595","type":"journal-article","created":{"date-parts":[[2021,12,2]],"date-time":"2021-12-02T16:07:00Z","timestamp":1638461220000},"page":"595","update-policy":"https:\/\/doi.org\/10.22331\/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":167,"title":["Almost-linear time decoding algorithm for topological codes"],"prefix":"10.22331","volume":"5","author":[{"given":"Nicolas","family":"Delfosse","sequence":"first","affiliation":[{"name":"IQIM, California Institute of Technology, Pasadena, CA, USA"},{"name":"Department of Physics and Astronomy, University of California, Riverside, CA, USA"},{"name":"Station Q Quantum Architectures and Computation Group, Microsoft Research, Redmond, WA 98052, USA"}]},{"given":"Naomi H.","family":"Nickerson","sequence":"additional","affiliation":[{"name":"Quantum Optics and Laser Science, Blackett Laboratory, Imperial College London, Prince Consort Road, London SW7 2AZ, United Kingdom"}]}],"member":"9598","published-online":{"date-parts":[[2021,12,2]]},"reference":[{"key":"0","doi-asserted-by":"publisher","unstructured":"Hussain Anwar, Benjamin J Brown, Earl T Campbell, and Dan E Browne. 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