Approximate Realization of Four-Dimensional Topological QuantumCodes on Two-Dimensional Architectures via Machine Learning andEntanglement-Assisted Floquet Engineering

Approximate Realization of Four-Dimensional Topological QuantumCodes on Two-Dimensional Architectures via Machine Learning andEntanglement-Assisted Floquet Engineering | 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 Approximate Realization of Four-Dimensional Topological QuantumCodes on Two-Dimensional Architectures via Machine Learning andEntanglement-Assisted Floquet Engineering Parham Ghayour This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-8331217/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 Four-dimensional (4D) topological quantum codes occupy a privileged position in quantum information theory: they are among the only known models exhibiting passive self-correction via macroscopic energy barriers and thermodynamic stability. However, physical quantum hardware is inherently limited to two or three spatial dimensions, making literal implementation of such codes impossible. In this work, we ask a different question: whether the logical and operational features of 4D topology can be approximated on two-dimensional (2D) hardware, even if the underlying phase of matter cannot. We introduce a framework for the approximate realization of 4D topological quantum codes on 2D architectures by combining three resources: synthetic dimensions induced by time-periodic control, entanglementassisted nonlocal connectivity, and machine-learning-based optimization. A mapping is constructed from the 4D toric code Hamiltonian to a periodically driven 2D system whose effective Floquet Hamiltonian reproduces the desired stabilizer structure up to controlled error. Virtual dimensions are realized using ancilla registers and time-indexed dynamics, while Bell-pair teleportation and graph-state primitives flatten geometric constraints and enable constant-depth realization of high-dimensional adjacency relations. Machine learning is deployed at three levels: (i) error-aware compilation and Floquet scheduling, (ii) learned stabilization and control pulse design, and (iii) neural decoding robust to mapping-induced distortions. We define precise operational metrics for approximation accuracy using channel distance, logical error rates, and effective barrier proxies, and provide rigorous error bounds from Trotter–Suzuki expansions and Floquet–Magnus theory. Numerical experiments demonstrate that entanglement-assisted routing suppresses geometric overhead, actionable neural decoding significantly reduces logical failure rates at finite size, and Floquet control yields tunable convergence toward ideal 4D dynamics. Although true thermodynamic self-correction remains forbidden in two dimensions, the resulting system exhibits an emergent regime of effective dimensional behavior in which key logical features of four-dimensional topology are operationally reproduced. Our results establish that dimensionality is not solely a geometric constraint but also a programmable resource mediated by entanglement, control, and inference. This work opens a pathway toward exploiting higher-dimensional code structure on near-term planar hardware and introduces a general paradigm for dimensionally enhanced quantum architectures. Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Under Review Version 1 posted Reviewers invited by journal 30 Apr, 2026 Editor assigned by journal 19 Feb, 2026 Submission checks completed at journal 10 Dec, 2025 First submitted to journal 10 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-8331217","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":634895284,"identity":"73971afc-3f23-47f2-911f-110b33a4165d","order_by":0,"name":"Parham Ghayour","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAABQ0lEQVRIie3RsWrCQBjA8YMrTpGuVyK5JxAuHASk4GN06XKh4C2RCoHi4BAQ4tJ2TsnQV6gIzoFAugRnIaXYxflEECepdw3FVDt063D/4XKB+91HCAA63X+NlU+oVvT1YsklWfyFUEXY8elKFeIGh7cd1Bw9pGLRT2+BOVyue4MpwNfDJVqF7/z5Kv3YT2lbzaBCnHzWQSxPW0Ejo3GUFcB+y5yLp9DvjosO2ZMb6iRVMvcc4IYpAYhRaNT2JGKOWQ9ZdxwzSRJ3ekSocHeS8DU0dpLwjSTcjrn4hRDkBpJ4FNbDAmDkqSkMm97pKXnuIJZxUmvkPqw/FgZBnt+KZsx+Mb1ewsjxt7ze09V2cEnOzdEEGpvCwhEfz8UdwzjmEyH6besH+a5W/g6DJOBM7Yk6SU4fV5UE4ABAUW50Op1OJ/sEKM97VF9810MAAAAASUVORK5CYII=","orcid":"","institution":"","correspondingAuthor":true,"prefix":"","firstName":"Parham","middleName":"","lastName":"Ghayour","suffix":""}],"badges":[],"createdAt":"2025-12-10 23:53:16","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-8331217/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-8331217/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":108807664,"identity":"aa9b002e-5188-4c4b-8694-df945bd5f6fa","added_by":"auto","created_at":"2026-05-08 15:31:06","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":400218,"visible":true,"origin":"","legend":"","description":"","filename":"4D.pdf","url":"https://assets-eu.researchsquare.com/files/rs-8331217/v1_covered_058f1b36-9cd7-48b8-8cca-57a9092a2722.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Approximate Realization of Four-Dimensional Topological QuantumCodes on Two-Dimensional Architectures via Machine Learning andEntanglement-Assisted Floquet Engineering","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":"","lastPublishedDoi":"10.21203/rs.3.rs-8331217/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-8331217/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eFour-dimensional (4D) topological quantum codes occupy a privileged position in quantum information theory: they are among the only known models exhibiting passive self-correction via macroscopic energy barriers and thermodynamic stability. However, physical quantum hardware is inherently limited to two or three spatial dimensions, making literal implementation of such codes impossible. In this work, we ask a different question: whether the logical and operational features of 4D topology can be approximated on two-dimensional (2D) hardware, even if the underlying phase of matter cannot.\u003c/p\u003e\n\u003cp\u003eWe introduce a framework for the approximate realization of 4D topological quantum codes on 2D architectures by combining three resources: synthetic dimensions induced by time-periodic control, entanglementassisted nonlocal connectivity, and machine-learning-based optimization. A mapping is constructed from the 4D toric code Hamiltonian to a periodically driven 2D system whose effective Floquet Hamiltonian reproduces the desired stabilizer structure up to controlled error. 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