{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,24]],"date-time":"2026-09-24T16:46:38Z","timestamp":1790268398741,"version":"4.1.0"},"reference-count":19,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2015,4,1]],"date-time":"2015-04-01T00:00:00Z","timestamp":1427846400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001711","name":"snsf","doi-asserted-by":"publisher","award":["nccr qsit"],"award-info":[{"award-number":["nccr qsit"]}],"id":[{"id":"10.13039\/501100001711","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Here we study an efficient algorithm for decoding topological codes. It is a simple form of HDRG decoder, which could be straightforwardly generalized to complex decoding problems. Specific results are obtained for the planar code with both i.i.d. and spatially correlated errors. The method is shown to compare well with existing ones, despite its simplicity.<\/jats:p>","DOI":"10.3390\/e17041946","type":"journal-article","created":{"date-parts":[[2015,4,7]],"date-time":"2015-04-07T03:47:46Z","timestamp":1428378466000},"page":"1946-1957","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":26,"title":["A Simple Decoder for Topological Codes"],"prefix":"10.3390","volume":"17","author":[{"given":"James","family":"Wootton","sequence":"first","affiliation":[{"name":"Department of Physics, University of Basel, Klingelbergstrasse 82, CH-4056 Basel, Switzerland"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2015,4,1]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1717","DOI":"10.1080\/09500340.2012.737937","article-title":"Quantum memories and error correction","volume":"20","author":"Wootton","year":"2012","journal-title":"J. Mod. Opt"},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Wootton, J.R., Burri, J., Iblisdir, S., and Loss, D. (2014). Error correction for non-Abelian topological quantum computation. Phys. Rev. X, 4.","DOI":"10.1103\/PhysRevX.4.011051"},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Bravyi, S., and Haah, J. (2013). Quantum self-correction in the 3D cubic code model. Phys. Rev. Lett, 111.","DOI":"10.1103\/PhysRevLett.111.200501"},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Anwar, H., Brown, B.J., Campbell, E.T., and Browne, D.E. (2014). Fast decoders for qudit topological codes. New J. Phys, 16.","DOI":"10.1088\/1367-2630\/16\/6\/063038"},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Hutter, A., Loss, D., and Wootton, J.R. (2014). Improved HDRG decoders for qudit and non-Abelian quantum error correction, arXiv:1410.4478.","DOI":"10.1088\/1367-2630\/17\/3\/035017"},{"key":"ref_6","unstructured":"Dennis, E. (2005). Purifying quantum states: Quantum and classical algorithms, arXiv:quant-ph\/0503169."},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Dennis, E., Kitaev, A., Landahl, A., and Preskill, J. (2002). Topological quantum memory. J. Math. Phys, 43.","DOI":"10.1063\/1.1499754"},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Hutter, A., Wootton, J.R., and Loss, D. (2014). Efficient Markov chain Monte Carlo algorithm for the surface code. Phys. Rev. A, 89.","DOI":"10.1103\/PhysRevA.89.022326"},{"key":"ref_9","unstructured":"Fowler, A.G. (2013). Optimal complexity correction of correlated errors in the surface code, arXiv:1310.0863."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Bravyi, S., Suchara, M., and Vargo, A. (2014). Efficient algorithms for maximum likelihood decoding in the surface code. Phys. Rev. A, 90.","DOI":"10.1103\/PhysRevA.90.032326"},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Novais, E., and Mucciolo, E.R. (2013). Surface code threshold in the presence of correlated errors. Phys. Rev. Lett, 110.","DOI":"10.1103\/PhysRevLett.110.010502"},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Hutter, A., and Loss, D. (2014). Breakdown of surface-code error correction due to coupling to a bosonic bath. Phys. Rev. A, 89.","DOI":"10.1103\/PhysRevA.89.042334"},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Fowler, A.G., and Martinis, J.M. (2014). Quantifying the effects of local many-qubit errors and nonlocal two-qubit errors on the surface code. Phys. Rev. A, 89.","DOI":"10.1103\/PhysRevA.89.032316"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"2","DOI":"10.1016\/S0003-4916(02)00018-0","article-title":"Fault-tolerant quantum computation by anyons","volume":"303","author":"Kitaev","year":"2003","journal-title":"Ann. Phys"},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Kovalev, A.A., and Pryadko, L.P. (2013). Fault tolerance of quantum low-density parity check codes with sublinear distance scaling. Phys. Rev. A, 87.","DOI":"10.1103\/PhysRevA.87.020304"},{"key":"ref_16","unstructured":"Landahl, A. (2013). (Albuquerque, NM, USA). Private communication."},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Duclos-Cianci, G., and Poulin, D. (2010). Fast decoders for topological quantum codes. Phys. Rev. Lett, 104.","DOI":"10.1103\/PhysRevLett.104.050504"},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Fowler, A.G. (2013). Accurate simulations of planar topological codes cannot use cyclic boundaries. Phys. Rev. A, 87.","DOI":"10.1103\/PhysRevA.87.062320"},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Bravyi, S., and Vargo, A. (2013). Simulation of rare events in quantum error correction. Phys. Rev. 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