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We analyze Markovian autonomous decoders that can be implemented with a wide range of qubit and bosonic error-correcting codes, and derive several upper bounds and a lower bound on the logical error rate in terms of correction and noise rates. These bounds suggest that, in general, there is always a correction rate, possibly size-dependent, above which autonomous memories exhibit arbitrarily long coherence times. For any given autonomous memory, size dependence of this correction rate is difficult to rule out: we point to common scenarios where autonomous decoders that stochastically implement active error correction must operate at rates that grow with code size. For codes with a threshold, we show that it is possible to achieve faster-than-polynomial decay of the logical error rate with code size by using superlogarithmic scaling of the correction rate. We illustrate our results with several examples. One example is an exactly solvable global dissipative toric code model that can achieve an effective logical error rate that decreases exponentially with the linear lattice size, provided that the recovery rate grows proportionally with the linear lattice size.<\/jats:p>","DOI":"10.22331\/q-2025-07-22-1804","type":"journal-article","created":{"date-parts":[[2025,7,22]],"date-time":"2025-07-22T08:46:28Z","timestamp":1753173988000},"page":"1804","update-policy":"https:\/\/doi.org\/10.22331\/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":3,"title":["Bounds on Autonomous Quantum Error Correction"],"prefix":"10.22331","volume":"9","author":[{"given":"Oles","family":"Shtanko","sequence":"first","affiliation":[{"name":"IBM Quantum, IBM Research \u2013 Almaden, San Jose, CA, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Yu-Jie","family":"Liu","sequence":"additional","affiliation":[{"name":"Technical University of Munich, TUM School of Natural Sciences, Physics Department, 85748 Garching, Germany"},{"name":"Munich Center for Quantum Science and Technology (MCQST), Munich, Germany"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Simon","family":"Lieu","sequence":"additional","affiliation":[{"name":"Joint Quantum Institute, NIST\/University of Maryland, College Park, MD, USA"},{"name":"Joint Center for Quantum Information and Computer Science, NIST\/University of Maryland, College Park, MD, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Alexey V.","family":"Gorshkov","sequence":"additional","affiliation":[{"name":"Joint Quantum Institute, NIST\/University of Maryland, College Park, MD, USA"},{"name":"Joint Center for Quantum Information and Computer Science, NIST\/University of Maryland, College Park, MD, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Victor V.","family":"Albert","sequence":"additional","affiliation":[{"name":"Joint Center for Quantum Information and Computer Science, NIST\/University of Maryland, College Park, MD, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"9598","published-online":{"date-parts":[[2025,7,22]]},"reference":[{"key":"0","doi-asserted-by":"publisher","unstructured":"Barbara M. 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