Quantum Error Correction under Full Noise using a Unified Geometric Equation

Quantum Error Correction under Full Noise using a Unified Geometric Equation | 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 Article Quantum Error Correction under Full Noise using a Unified Geometric Equation Hisham Baroudi This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-6753266/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 present a unified equation for modeling quantum error correction that connects logical fidelity, error amplitude, and the intrinsic geometry of quantum systems. The total error E is defined as: E = ρ * (1 + ε) + γ * π In this expression: ● ρ (rho) represents the logical fidelity after the correction is applied. ● ε (epsilon) is the magnitude of the introduced error. ● γ (gamma) is a coupling factor for phase-based influence. ● π (pi) reflects the irreducible geometric curvature of the quantum phase space. To test this equation, we simulate a three-qubit repetition code using Cirq, including full depolarizing noise (10%) applied to all qubits. The results demonstrate that this equation remains valid even under realistic noisy conditions. Even when the fidelity is high, the geometric term γ * π reveals a base-level error that cannot be corrected — a limit set by the system’s geometry. This model offers a predictive framework for understanding quantum error, going beyond operational recovery to include structural constraints. Physical sciences/Physics/Quantum physics/Quantum mechanics Physical sciences/Physics/Quantum physics/Quantum information Full Text Additional Declarations There is NO Competing Interest. 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. Our growing team is made up of researchers and industry professionals working together to solve the most critical problems facing scientific publishing. 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-6753266","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Article","associatedPublications":[],"authors":[{"id":462907709,"identity":"45f5fc4d-4894-49e8-8a26-a17780e0fbc3","order_by":0,"name":"Hisham Baroudi","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA1ElEQVRIiWNgGAWjYBACAwYeMM3Yxt4DZvDwEa+F5wwDwwGgFjaitTRI5IC1MBDUYs7Ae/BzRcVh2T7Jtwcff8yxk2FjYH746AYeLZYNfMmSZ84cNm6Tzks2OLgtGegwNmPjHHwOu//GQLKx7XBim3SOmcTBbcxALTxs0ni1HOAx/gnWInkGpKWeKC1mEFskeEBaDhOjhS/NsuFMunEbT46xwdltx3nYmAn55QDv4ZsNFday89vPGD6o3FZtz8/e/PAxPi1YADNpykfBKBgFo2AUYAEAcepFInLFSWAAAAAASUVORK5CYII=","orcid":"","institution":"CEC","correspondingAuthor":true,"prefix":"","firstName":"Hisham","middleName":"","lastName":"Baroudi","suffix":""}],"badges":[],"createdAt":"2025-05-26 18:45:20","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-6753266/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-6753266/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":84213990,"identity":"84e6b05d-65db-474d-9b37-5f15f67aa741","added_by":"auto","created_at":"2025-06-09 10:24:16","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":305571,"visible":true,"origin":"","legend":"Article File","description":"","filename":"COPYOF1.pdf","url":"https://assets-eu.researchsquare.com/files/rs-6753266/v1_covered_829ebc92-5e15-4c41-9500-71fcd6a14730.pdf"}],"financialInterests":"There is \u003cb\u003eNO\u003c/b\u003e Competing Interest.","formattedTitle":"Quantum Error Correction under Full Noise using a Unified Geometric Equation","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":"","lastPublishedDoi":"10.21203/rs.3.rs-6753266/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-6753266/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eWe present a unified equation for modeling quantum error correction that connects logical fidelity, error amplitude, and the intrinsic geometry of quantum systems. The total error E is defined as:\u003c/p\u003e\n\u003cp\u003eE = ρ * (1 + ε) + γ * π\u003c/p\u003e\n\u003cp\u003eIn this expression:\u003c/p\u003e\n\u003cp\u003e● ρ (rho) represents the logical fidelity after the correction is applied.\u003c/p\u003e\n\u003cp\u003e● ε (epsilon) is the magnitude of the introduced error.\u003c/p\u003e\n\u003cp\u003e● γ (gamma) is a coupling factor for phase-based influence.\u003c/p\u003e\n\u003cp\u003e● π (pi) reflects the irreducible geometric curvature of the quantum phase space.\u003c/p\u003e\n\u003cp\u003eTo test this equation, we simulate a three-qubit repetition code using Cirq, including full depolarizing noise (10%) applied to all qubits. The results demonstrate that this equation remains valid even under realistic noisy conditions. Even when the fidelity is high, the geometric term γ * π reveals a base-level error that cannot be corrected — a limit set by the system’s geometry. 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