Efficient Reconciliation of Continuous Variable Quantum Key Distribution with Multiplicatively Repeated Non-Binary LDPC Codes

Efficient Reconciliation of Continuous Variable Quantum Key Distribution with Multiplicatively Repeated Non-Binary LDPC 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 Article Efficient Reconciliation of Continuous Variable Quantum Key Distribution with Multiplicatively Repeated Non-Binary LDPC Codes Jesus Martinez-Mateo, David Elkouss This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-5922750/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 Continuous variable quantum key distribution bears the promise of simple quantum key distribution directly compatible with commercial off the shelf equipment. However, for a long time its performance was hindered by the absence of good classical postprocessing capable of distilling secret-keys in the noisy regime. Advanced coding solutions in the past years have partially addressed this problem enabling record transmission distances of up to 165 km, and 206 km over ultra-low loss fiber. In this paper, we show that a very simple coding solution with a single code is sufficient to extract keys at all noise levels. This solution has performance competitive with prior results for all levels of noise, and we show that non-zero keys can be distilled up to a record distance of 192 km assuming the standard loss of a single-mode optical fiber, and 240 km over ultra-low loss fibers. Low-rate codes are constructed using multiplicatively repeated non-binary low-density parity-check codes over a finite field of characteristic two. This construction only makes use of a $(2, k)$-regular non-binary low-density parity-check code as mother code, such that code design is in fact not required, thus trivializing the code construction procedure. The construction is also inherently rate-adaptive thereby allowing to easily create codes of any rate. Rate-adaptive codes are of special interest for the efficient reconciliation of errors over time or arbitrary varying channels, as is the case with quantum key distribution. In short, these codes are highly efficient when reconciling errors over a very noisy communication channel, and perform well even for short block-length codes. Finally, the proposed solution is known to be easily amenable to hardware implementations, thus addressing also the requirements for practical reconciliation in continuous variable quantum key distribution. Physical sciences/Physics/Information theory and computation Physical sciences/Physics/Quantum physics/Quantum information Physical sciences/Mathematics and computing/Computer science Full Text Additional Declarations No competing interests reported. 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-5922750","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Article","associatedPublications":[],"authors":[{"id":409575333,"identity":"91e7d161-7880-43ef-85c3-f2320cd87de0","order_by":0,"name":"Jesus Martinez-Mateo","email":"","orcid":"","institution":"Technical University of Madrid","correspondingAuthor":false,"prefix":"","firstName":"Jesus","middleName":"","lastName":"Martinez-Mateo","suffix":""},{"id":409575336,"identity":"9e4c14cd-cb65-4047-8c2f-471449031e39","order_by":1,"name":"David Elkouss","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA80lEQVRIie3QMQrCMBSA4VcEp6dzSsVeIRJwKnoVRejUQRDEwaFQqIsHcCh6CME5EohLwdVdcHLQTSdNVQQdUt1E8hOaZPjgpQAm0y/G1UKACrndBoDZRj8heCepIsUPCDyJFatPMWeu8iqt8T14aI/G7HieehV3FkKvqyF2GtBlAj46mNad0sJHKgHYREMo91sCQWCVBPWCtRBI1WAMdWS9y8glI2qwRKAb55FNhyvC0SEBJaVQnWUOsTdbvkxoB+2x7Dsos7e0Q+1byut2dNgPGlWyiubH09BrupGQTPfHHuO93KyY5Yr3CtuviclkMv1zV0E6Rzvu5nRHAAAAAElFTkSuQmCC","orcid":"","institution":"Okinawa Institute of Science and Technology","correspondingAuthor":true,"prefix":"","firstName":"David","middleName":"","lastName":"Elkouss","suffix":""}],"badges":[],"createdAt":"2025-01-29 08:23:12","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-5922750/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-5922750/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":76370908,"identity":"7951392f-3b6d-499c-a83c-7ed4ba750785","added_by":"auto","created_at":"2025-02-15 18:16:32","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":693767,"visible":true,"origin":"","legend":"","description":"","filename":"LowrateQKD.pdf","url":"https://assets-eu.researchsquare.com/files/rs-5922750/v1_covered_4f0d92bc-8345-4b52-9ee9-267ba33323fb.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Efficient Reconciliation of Continuous Variable Quantum Key Distribution with Multiplicatively Repeated Non-Binary LDPC Codes","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"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-5922750/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-5922750/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"Continuous variable quantum key distribution bears the promise of simple quantum key distribution directly compatible with commercial off the shelf equipment. However, for a long time its performance was hindered by the absence of good classical postprocessing capable of distilling secret-keys in the noisy regime. Advanced coding solutions in the past years have partially addressed this problem enabling record transmission distances of up to 165~km, and 206~km over ultra-low loss fiber. In this paper, we show that a very simple coding solution with a single code is sufficient to extract keys at all noise levels. This solution has performance competitive with prior results for all levels of noise, and we show that non-zero keys can be distilled up to a record distance of 192~km assuming the standard loss of a single-mode optical fiber, and 240~km over ultra-low loss fibers. Low-rate codes are constructed using multiplicatively repeated non-binary low-density parity-check codes over a finite field of characteristic two. This construction only makes use of a $(2, k)$-regular non-binary low-density parity-check code as mother code, such that code design is in fact not required, thus trivializing the code construction procedure. The construction is also inherently rate-adaptive thereby allowing to easily create codes of any rate. Rate-adaptive codes are of special interest for the efficient reconciliation of errors over time or arbitrary varying channels, as is the case with quantum key distribution. In short, these codes are highly efficient when reconciling errors over a very noisy communication channel, and perform well even for short block-length codes. 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