Quantum-Classical Hybrid Quantized Neural Network

Quantum-Classical Hybrid Quantized Neural Network | 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-Classical Hybrid Quantized Neural Network Chuan Wang, Wen-Xin Li, Hong-Dong Zhu, Qi Gao, Yin Ma, Hai Wei, and 1 more This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-6725857/v1 This work is licensed under a CC BY 4.0 License Status: Under Review Version 1 posted You are reading this latest preprint version Abstract Here in this work, we present a novel Quadratic Binary Optimization (QBO) model for quantized neural network training, enabling the use of arbitrary activation and loss functions through spline interpolation. We introduce Forward Interval Propagation (FIP), a method designed to tackle the challenges of non-linearity and the multi-layer composite structure in neural networks by discretizing activation functions into linear subintervals. This approach preserves the universal approximation properties of neural networks while allowing complex nonlinear functions to be optimized using quantum computers, thus broadening their applicability in artificial intelligence. We provide theoretical upper bounds on the approximation error and the number of Ising spins required, which is independent of the training sample size. A significant challenge in solving the associated Quadratic Constrained Binary Optimization (QCBO) model on a large scale is the presence of numerous constraints. When employing the penalty method to handle these constraints, tuning a large number of penalty coefficients becomes a critical hyperparameter optimization problem, increasing computational complexity and potentially affecting solution quality. To address this, we employ the Quantum Conditional Gradient Descent (QCGD) algorithm, which leverages quantum computing to directly solve the QCBO problem. We prove the convergence of QCGD under a quantum oracle with randomness and bounded variance in objective value, as well as under limited precision constraints in the coefficient matrix. Additionally, we provide an upper bound on the Time-To-Solution for the QCBO solving process. Experimental results using a coherent Ising machine (CIM) demonstrate a $94.95%$ accuracy on the Fashion MNIST classification task, with comparisons to classical algorithms highlighting superior efficiency. We further validate the robustness of the spline interpolation method and analyze the impact of varying precision levels on QCGD convergence, underscoring the reliability of our framework. Physical sciences/Mathematics and computing/Computer science Physical sciences/Optics and photonics/Optical physics/Quantum optics Full Text Additional Declarations There is NO Competing Interest. Supplementary Files neuralnetworkQUBOsupplementary.pdf Supplementary Materials of Quantum-Classical Hybrid Quantized Neural Network Cite Share Download PDF Status: Under Review 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-6725857","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Article","associatedPublications":[],"authors":[{"id":467666442,"identity":"f72a4287-fbb1-4daf-a1da-e5d70188eac0","order_by":0,"name":"Chuan Wang","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA10lEQVRIiWNgGAWjYLACxgYgwd6DYBOphecMyVokcojUYnD87OEXP3ccljeXfHvsMQ+DjeyGA8zPHuDVciYvzbL3zGHDnbPz0o15GNKMNxxgMzfAp8XsQI6ZMWPbYcYNt3PMpHkYDiduOMDDJoFXy/k3YC32G26eAWn5T4SWGznGj4FaEjfc4AFpOUBYi/2NN2aMvW3pyRuAnpKcY5BsPPMwmxleLZL9OcYffrZZ2244fvaYxJsKO9m+483P8GoBAmRngIKKmYB6kJIPhNWMglEwCkbBiAYA6xZMCU4dL+cAAAAASUVORK5CYII=","orcid":"","institution":"Beijing Normal University","correspondingAuthor":true,"prefix":"","firstName":"Chuan","middleName":"","lastName":"Wang","suffix":""},{"id":467666443,"identity":"b0c89065-6e1c-4ded-b343-3cc8fccfd90f","order_by":1,"name":"Wen-Xin Li","email":"","orcid":"https://orcid.org/0009-0006-8114-9254","institution":"Beijing QBoson Quantum Technology Co., Ltd.","correspondingAuthor":false,"prefix":"","firstName":"Wen-Xin","middleName":"","lastName":"Li","suffix":""},{"id":467666444,"identity":"0b52d8d3-03e4-465f-b154-ae79dec92c73","order_by":2,"name":"Hong-Dong Zhu","email":"","orcid":"","institution":"Beijing QBoson Quantum Technology Co., Ltd.","correspondingAuthor":false,"prefix":"","firstName":"Hong-Dong","middleName":"","lastName":"Zhu","suffix":""},{"id":467666445,"identity":"a649df5d-9a6e-43d7-9ef0-8965798feb7b","order_by":3,"name":"Qi Gao","email":"","orcid":"","institution":"Beijing QBoson Quantum Technology Co., Ltd.","correspondingAuthor":false,"prefix":"","firstName":"Qi","middleName":"","lastName":"Gao","suffix":""},{"id":467666446,"identity":"74a40ec2-4256-446f-b8e4-f1ae39c17144","order_by":4,"name":"Yin Ma","email":"","orcid":"","institution":"Beijing QBoson Quantum Technology Co., Ltd.","correspondingAuthor":false,"prefix":"","firstName":"Yin","middleName":"","lastName":"Ma","suffix":""},{"id":467666447,"identity":"db4a8b4c-f249-43d8-8e8e-af36d31a3508","order_by":5,"name":"Hai Wei","email":"","orcid":"https://orcid.org/0009-0003-8382-8283","institution":"Beijing QBoson Quantum Technology Co., Ltd.","correspondingAuthor":false,"prefix":"","firstName":"Hai","middleName":"","lastName":"Wei","suffix":""},{"id":467666448,"identity":"1f091ca3-24cc-48c4-aee1-ec2a85de6597","order_by":6,"name":"Kai Wen","email":"","orcid":"https://orcid.org/0000-0002-7496-1248","institution":"Beijing QBoson Quantum Technology Co., Ltd.","correspondingAuthor":false,"prefix":"","firstName":"Kai","middleName":"","lastName":"Wen","suffix":""}],"badges":[],"createdAt":"2025-05-22 14:15:47","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-6725857/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-6725857/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":86131019,"identity":"134fdced-7afe-41ff-9818-7666524da6dc","added_by":"auto","created_at":"2025-07-07 06:44:16","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":6808212,"visible":true,"origin":"","legend":"Article File","description":"","filename":"neuralnetworkQUBOmanuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-6725857/v1_covered_165ff8e4-0836-4528-93c0-de7c0e353b83.pdf"},{"id":86129952,"identity":"82792887-dab0-451a-83ee-b820e9a2dbff","added_by":"auto","created_at":"2025-07-07 06:35:59","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"supplement","size":782930,"visible":true,"origin":"","legend":"Supplementary Materials of Quantum-Classical Hybrid Quantized Neural Network","description":"","filename":"neuralnetworkQUBOsupplementary.pdf","url":"https://assets-eu.researchsquare.com/files/rs-6725857/v1/73ed8694cb71a8637d56b6b6.pdf"}],"financialInterests":"There is \u003cb\u003eNO\u003c/b\u003e Competing Interest.","formattedTitle":"Quantum-Classical Hybrid Quantized Neural Network","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":true,"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":"nature-portfolio","isNatureJournal":true,"hasQc":false,"allowDirectSubmit":false,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"","title":"Nature Portfolio","twitterHandle":"","acdcEnabled":false,"dfaEnabled":false,"editorialSystem":"ejp","reportingPortfolio":"","inReviewEnabled":true,"inReviewRevisionsEnabled":false},"keywords":"","lastPublishedDoi":"10.21203/rs.3.rs-6725857/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-6725857/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"Here in this work, we present a novel Quadratic Binary Optimization (QBO) model for quantized neural network training, enabling the use of arbitrary activation and loss functions through spline interpolation. We introduce Forward Interval Propagation (FIP), a method designed to tackle the challenges of non-linearity and the multi-layer composite structure in neural networks by discretizing activation functions into linear subintervals. This approach preserves the universal approximation properties of neural networks while allowing complex nonlinear functions to be optimized using quantum computers, thus broadening their applicability in artificial intelligence. We provide theoretical upper bounds on the approximation error and the number of Ising spins required, which is independent of the training sample size. A significant challenge in solving the associated Quadratic Constrained Binary Optimization (QCBO) model on a large scale is the presence of numerous constraints. When employing the penalty method to handle these constraints, tuning a large number of penalty coefficients becomes a critical hyperparameter optimization problem, increasing computational complexity and potentially affecting solution quality. To address this, we employ the Quantum Conditional Gradient Descent (QCGD) algorithm, which leverages quantum computing to directly solve the QCBO problem. We prove the convergence of QCGD under a quantum oracle with randomness and bounded variance in objective value, as well as under limited precision constraints in the coefficient matrix. Additionally, we provide an upper bound on the Time-To-Solution for the QCBO solving process. Experimental results using a coherent Ising machine (CIM) demonstrate a $94.95\\%$ accuracy on the Fashion MNIST classification task, with comparisons to classical algorithms highlighting superior efficiency. We further validate the robustness of the spline interpolation method and analyze the impact of varying precision levels on QCGD convergence, underscoring the reliability of our framework.","manuscriptTitle":"Quantum-Classical Hybrid Quantized Neural Network","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2025-07-07 06:35:54","doi":"10.21203/rs.3.rs-6725857/v1","editorialEvents":[],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"nature-communications","isNatureJournal":true,"hasQc":false,"allowDirectSubmit":false,"externalIdentity":"NCOMMS","sideBox":"Learn more about [Nature Communications](http://www.nature.com/ncomms/)","snPcode":"","submissionUrl":"https://mts-ncomms.nature.com/","title":"Nature Communications","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"ejp","reportingPortfolio":"Nature Communications","inReviewEnabled":true,"inReviewRevisionsEnabled":false}}],"origin":"","ownerIdentity":"1e8ed750-54dc-44e4-820e-c13830628343","owner":[],"postedDate":"July 7th, 2025","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"under-review","subjectAreas":[{"id":49659293,"name":"Physical sciences/Mathematics and computing/Computer science"},{"id":49659294,"name":"Physical sciences/Optics and photonics/Optical physics/Quantum optics"}],"tags":[],"updatedAt":"2025-12-07T00:00:11+00:00","versionOfRecord":[],"versionCreatedAt":"2025-07-07 06:35:54","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-6725857","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-6725857","identity":"rs-6725857","version":["v1"]},"buildId":"8U1c8b4HqxoKbykW_rLl7","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}