NQ-SVM: Scalable Quantum Kernel Learning via Nyström-Based Support Vector Machines

NQ-SVM: Scalable Quantum Kernel Learning via Nyström-Based Support Vector Machines | 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 NQ-SVM: Scalable Quantum Kernel Learning via Nyström-Based Support Vector Machines Duong The Do, Srikanth Thudumu, Prem Prakash Jayaraman, Duong Tung Nguyen This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-8962794/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 Support vector machines (SVMs) are widely used for classification, but become computationally expensive as datasets grow and data become high-dimensional or nonlinearly separable. Quantum support vector machines (QSVMs) enhance the traditional SVMs by replacing conventional feature mappings with quantum-implemented embeddings that project data into high-dimensional Hilbert spaces, potentially improving expressive power and efficiency. However, training QSVMs on current noisy intermediate-scale quantum (NISQ) devices remains challenging, as constructing the quantum kernel matrix requires repeated circuit executions and scales quadratically with dataset size. We propose NQ-SVM , a hybrid quantum-classical method that incorporates the Nystr\"om approximation to enhance the scalability of QSVMs. By selecting a small set of landmark points, the Nystr\"om method yields a low-rank kernel approximation, reducing complexity from \((O(n^2))\) to \((O(nm))\) , while preserving the quantum advantages in handling high-dimensional, non-linear data. Extensive simulations demonstrate that NQ-SVM achieves comparable classification accuracy to classical SVMs and full QSVMs, while substantially reducing training time and kernel evaluation costs. These results underscore the potential of NQ-SVM as a scalable and practical hybrid framework for near-term quantum machine learning (QML). Quantum support vector machines Nystr "om approximation hybrid quantum-classical learning 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. 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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-8962794","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":596691393,"identity":"d7c286d3-4ecc-401a-9d7d-85c6c5164588","order_by":0,"name":"Duong The Do","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAs0lEQVRIiWNgGAWjYDCCAyCCjUEOwmMjQYsx6VoSG4jWwncj+djDL2U26f39ZwwYPpQdJqxF8kZaurHMubTcGTdyDBhnnCNCi8GNHDNpybbDuQ03eAyYedtI0JIuf/6MAfNfYrVIfmw7nGBwIMeAmZEYLZJnnqVJM5xLM9x4I63gYM+5dMJa+I4nH5P8UWYjL3f+8MYHP8qsCWsBAWYeKOMAceqBgPEH0UpHwSgYBaNgRAIALN49V49+NVsAAAAASUVORK5CYII=","orcid":"","institution":"Arizona State University","correspondingAuthor":true,"prefix":"","firstName":"Duong","middleName":"The","lastName":"Do","suffix":""},{"id":596691394,"identity":"91a6f3b7-1adc-4362-ad98-2be24eb73005","order_by":1,"name":"Srikanth Thudumu","email":"","orcid":"","institution":"Institute of Applied Artificial Intelligence and Robotics","correspondingAuthor":false,"prefix":"","firstName":"Srikanth","middleName":"","lastName":"Thudumu","suffix":""},{"id":596691395,"identity":"c66ceee3-d1ad-4d2c-a0ff-bcb7a11c05b9","order_by":2,"name":"Prem Prakash Jayaraman","email":"","orcid":"","institution":"Swinburne University of Technology","correspondingAuthor":false,"prefix":"","firstName":"Prem","middleName":"Prakash","lastName":"Jayaraman","suffix":""},{"id":596691396,"identity":"a5606a77-2557-47fe-908c-a323d62c7edc","order_by":3,"name":"Duong Tung Nguyen","email":"","orcid":"","institution":"Arizona State University","correspondingAuthor":false,"prefix":"","firstName":"Duong","middleName":"Tung","lastName":"Nguyen","suffix":""}],"badges":[],"createdAt":"2026-02-25 03:54:19","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-8962794/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-8962794/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":104401097,"identity":"777f16ca-1e6a-4c30-98d1-258e4e6127b0","added_by":"auto","created_at":"2026-03-11 12:11:51","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":3614795,"visible":true,"origin":"","legend":"","description":"","filename":"QNSVMQMISubmission.pdf","url":"https://assets-eu.researchsquare.com/files/rs-8962794/v1_covered_5d2d1364-5a3e-4d74-baff-0dab8435b2fd.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"NQ-SVM: Scalable Quantum Kernel Learning via Nyström-Based Support Vector Machines","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 support vector machines, Nystr, \"om approximation, hybrid quantum-classical learning","lastPublishedDoi":"10.21203/rs.3.rs-8962794/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-8962794/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eSupport vector machines (SVMs) are widely used for classification, but become computationally expensive as datasets grow and data become high-dimensional or nonlinearly separable. Quantum support vector machines (QSVMs) enhance the traditional SVMs by replacing conventional feature mappings with quantum-implemented embeddings that project data into high-dimensional Hilbert spaces, potentially improving expressive power and efficiency. However, training QSVMs on current noisy intermediate-scale quantum (NISQ) devices remains challenging, as constructing the quantum kernel matrix requires repeated circuit executions and scales quadratically with dataset size. We propose \u003cem\u003eNQ-SVM\u003c/em\u003e, a hybrid quantum-classical method that incorporates the Nystr\\\"om approximation to enhance the scalability of QSVMs. By selecting a small set of landmark points, the Nystr\\\"om method yields a low-rank kernel approximation, reducing complexity from \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\((O(n^2))\\)\u003c/span\u003e\u003c/span\u003e to \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\((O(nm))\\)\u003c/span\u003e\u003c/span\u003e, while preserving the quantum advantages in handling high-dimensional, non-linear data. Extensive simulations demonstrate that \u003cem\u003eNQ-SVM\u003c/em\u003e achieves comparable classification accuracy to classical SVMs and full QSVMs, while substantially reducing training time and kernel evaluation costs. 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