Finite-Size Decoding Guarantees for Stabilizer Quantum Error-Correcting Codes

Finite-Size Decoding Guarantees for Stabilizer Quantum Error-Correcting Codes | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Finite-Size Decoding Guarantees for Stabilizer Quantum Error-Correcting Codes Parham Ghayour This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-8391899/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract We develop a general framework for bounding logical error probabilities at finite code distance in stabilizer quantum error-correcting codes. Rather than relying on asymptotic threshold argu ments, the analysis isolates decoder locality and control over syndrome connectivity as sufficient conditions for exponential suppression of logical errors at finite size. Under a stochastic Pauli noise model, we show that component-local decoders admit an explicit finite-size bound determined by the probability of large connected syndrome structures. The framework is formulated for stabilizer codes defined on bounded-degree syndrome graphs and is independent of any specific decoding algorithm. We prove a general finite-size decoding theorem and illustrate its application to the two-dimensional surface code, where the bound scales exponentially with the code distance. We further discuss the extension of the framework to quantum low-density parity-check and hypergraph-product codes. A minimal numerical illustration confirms the qualitative finite-size scaling predicted by the theory. These results provide a complementary perspective to fault-tolerance threshold theorems by mak ing explicit the mechanisms governing logical error suppression at finite code distance, with direct relevance to near- and intermediate-scale quantum devices. quantum error correction fault tolerant quantum computation Full Text Additional Declarations The authors declare no competing interests. Cite Share Download PDF Status: Posted Version 1 posted You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. 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Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-8391899","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":562179061,"identity":"d2485c05-889b-46e4-9c9e-577cc9ff311b","order_by":0,"name":"Parham Ghayour","email":"data:image/png;base64,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","orcid":"https://orcid.org/0009-0001-5607-1532","institution":"Sorbonne university","correspondingAuthor":true,"prefix":"","firstName":"Parham","middleName":"","lastName":"Ghayour","suffix":""}],"badges":[],"createdAt":"2025-12-18 06:44:43","currentVersionCode":1,"declarations":{"humanSubjects":false,"vertebrateSubjects":false,"conflictsOfInterestStatement":false,"humanSubjectEthicalGuidelines":false,"humanSubjectConsent":false,"humanSubjectClinicalTrial":false,"humanSubjectCaseReport":false,"vertebrateSubjectEthicalGuidelines":false},"doi":"10.21203/rs.3.rs-8391899/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-8391899/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":98627124,"identity":"76cbb643-fb6b-4bc8-9dbf-ab6e184de488","added_by":"auto","created_at":"2025-12-19 17:10:09","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":276394,"visible":true,"origin":"","legend":"","description":"","filename":"QECrev2.pdf","url":"https://assets-eu.researchsquare.com/files/rs-8391899/v1_covered_e62929ed-5efc-4070-8f1b-078095a11f92.pdf"}],"financialInterests":"The authors declare no competing interests.","formattedTitle":"\u003cp\u003eFinite-Size Decoding Guarantees for Stabilizer Quantum Error-Correcting Codes\u003c/p\u003e","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":true,"hideJournal":true,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":true,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":true,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"quantum error correction, fault tolerant quantum computation","lastPublishedDoi":"10.21203/rs.3.rs-8391899/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-8391899/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eWe develop a general framework for bounding logical error probabilities at finite code distance in stabilizer quantum error-correcting codes. 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