Fusion of classical and quantum kernels enables accurate and robust two-sample tests

Fusion of classical and quantum kernels enables accurate and robust two-sample tests | 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 Fusion of classical and quantum kernels enables accurate and robust two-sample tests Yu Terada, Yugo Ogio, Ken Arai, Hiroyuki Tezuka, Yu Tanaka This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-8288303/v1 This work is licensed under a CC BY 4.0 License Status: Under Review Version 1 posted 4 You are reading this latest preprint version Abstract Two-sample tests have been extensively employed in various scientific fields and machine learning such as evaluation on the effectiveness of drugs and A/B testing on different marketing strategies to discriminate whether two sets of samples come from the same distribution or not. Kernel-based procedures for hypothetical testing have been proposed to efficiently disentangle high-dimensional complex structures in data to obtain accurate results in a model-free way by embedding the data into the reproducing kernel Hilbert space (RKHS). While the choice of kernels plays a crucial role for their performance, little is understood about how to choose kernel especially for small datasets. Here we aim to construct a hypothetical test which is effective even for small datasets, based on the theoretical foundation of kernel-based tests using maximum mean discrepancy, which is called MMD-FUSE. To address this, we enhance the MMD-FUSE framework by incorporating quantum kernels and propose a novel hybrid testing strategy that fuses classical and quantum kernels. This approach creates a powerful and adaptive test by combining the domain-specific inductive biases of classical kernels with the unique expressive power of quantum kernels. We evaluate our method on various synthetic and real-world clinical datasets, and our experiments reveal two key findings: 1) With appropriate hyperparameter tuning, MMD-FUSE with quantum kernels consistently improves test power over classical counterparts, especially for small and high-dimensional data. 2) The proposed hybrid framework demonstrates remarkable robustness, adapting to different data characteristics and achieving high test power across diverse scenarios. These results highlight the potential of quantum-inspired and hybrid kernel strategies to build more effective statistical tests, offering a versatile tool for data analysis where sample sizes are limited. two-sample test hypothetical test kernel choice quantum kernel Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Under Review Version 1 posted Reviewers invited by journal 23 Mar, 2026 Editor assigned by journal 19 Feb, 2026 Submission checks completed at journal 07 Dec, 2025 First submitted to journal 05 Dec, 2025 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-8288303","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":611212058,"identity":"efe33381-8832-4002-a93a-c04f3c59bb95","order_by":0,"name":"Yu Terada","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA5klEQVRIiWNgGAWjYBADxgYQ+QGI2diJVA0hZoC0MJOihZkHxCekxeBGAvuDj3vsZPsbmB9+tvm1TZ6PmYHxw8ccvFoYG2c8SzaecYDNWDq377ZhGzMDs+TMbfi1NPMcYE5sOMBgxpzbc5sRqIWNmZeQlj8H6hPnH2D/xmzZc9ueOC0MBw4nbjjAY8bM8ON2IkEtkmceNs7sOXDceONhnmLJ3obbyW3MjM14/cJ3PPnAhx8HqmXnHW/f+OHHn9u289ubD374iEeLwgFIvEOig7ENxIKJ4ADyqNJ/8CoeBaNgFIyCEQoAf4ZVNr9PFfsAAAAASUVORK5CYII=","orcid":"","institution":"Sony Group Corporation","correspondingAuthor":true,"prefix":"","firstName":"Yu","middleName":"","lastName":"Terada","suffix":""},{"id":611212059,"identity":"5d3498a4-b6cd-4b27-a9dd-c1aac143d85b","order_by":1,"name":"Yugo Ogio","email":"","orcid":"","institution":"Sony Group Corporation","correspondingAuthor":false,"prefix":"","firstName":"Yugo","middleName":"","lastName":"Ogio","suffix":""},{"id":611212060,"identity":"081e0d0a-10c6-46e6-91e1-2ec70d1167ce","order_by":2,"name":"Ken Arai","email":"","orcid":"","institution":"Sony Group Corporation","correspondingAuthor":false,"prefix":"","firstName":"Ken","middleName":"","lastName":"Arai","suffix":""},{"id":611212061,"identity":"6845cb29-278f-40c6-8625-42e3cbb343b6","order_by":3,"name":"Hiroyuki Tezuka","email":"","orcid":"","institution":"Sony Group Corporation","correspondingAuthor":false,"prefix":"","firstName":"Hiroyuki","middleName":"","lastName":"Tezuka","suffix":""},{"id":611212062,"identity":"e597fa2f-b767-4b5b-a165-0b6467845fe4","order_by":4,"name":"Yu Tanaka","email":"","orcid":"","institution":"Sony Group Corporation","correspondingAuthor":false,"prefix":"","firstName":"Yu","middleName":"","lastName":"Tanaka","suffix":""}],"badges":[],"createdAt":"2025-12-05 13:38:13","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-8288303/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-8288303/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":105562963,"identity":"92d2dcdf-a09a-461c-a80d-6f16b796a494","added_by":"auto","created_at":"2026-03-27 12:45:26","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":558620,"visible":true,"origin":"","legend":"","description":"","filename":"journalsubmission.pdf","url":"https://assets-eu.researchsquare.com/files/rs-8288303/v1_covered_2ae39ac1-c84c-4d8e-a8ef-53617c46f697.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Fusion of classical and quantum kernels enables accurate and robust two-sample tests","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":false,"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":"quantum-machine-intelligence","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"qumi","sideBox":"Learn more about [Quantum Machine Intelligence](http://link.springer.com/journal/42484)","snPcode":"42484","submissionUrl":"https://submission.nature.com/new-submission/42484/3","title":"Quantum Machine Intelligence","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"em","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false},"keywords":"two-sample test, hypothetical test, kernel choice, quantum kernel","lastPublishedDoi":"10.21203/rs.3.rs-8288303/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-8288303/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"Two-sample tests have been extensively employed in various scientific fields and machine learning such as evaluation on the effectiveness of drugs and A/B testing on different marketing strategies to discriminate whether two sets of samples come from the same distribution or not. 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