Extremal Magic States from Symmetric Lattices

Extremal Magic States from Symmetric Lattices | 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Research Article Extremal Magic States from Symmetric Lattices Misaki Ohta, Kazuki Sakurai This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-7784331/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 Magic, a key quantum resource beyond entanglement, remains poorly understood in terms of its structure and classification. In this paper, we demonstrate a striking connection between high-dimensional symmetric lattices and quantum magic states. By mapping vectors from the $E_8, B W_{16}$, and $E_6$ lattices into Hilbert space, we construct and classify stabiliser and maximal magic states for two-qubit, three-qubit and one-qutrit systems. In particular, this geometric approach allows us to construct, for the first time, closed-form expressions for the maximal magic states in the three-qubit and one-qutrit systems, and to conjecture their total counts. In the three-qubit case, we further classify the extremal magic states according to their entanglement structure. We also examine the distinctive behaviour of one-qutrit maximal magic states with respect to Clifford orbits. Our findings suggest that deep algebraic and geometric symmetries underlie the structure of extremal magic states. Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Under Review Version 1 posted Reviewers invited by journal 16 Jan, 2026 Editor assigned by journal 10 Oct, 2025 Submission checks completed at journal 10 Oct, 2025 First submitted to journal 05 Oct, 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-7784331","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":576304441,"identity":"592d3bb1-73d3-4842-843c-8751b72abf26","order_by":0,"name":"Misaki Ohta","email":"data:image/png;base64,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","orcid":"","institution":"University of Wrocław","correspondingAuthor":true,"prefix":"","firstName":"Misaki","middleName":"","lastName":"Ohta","suffix":""},{"id":576304445,"identity":"d0f807f1-db7f-4cd0-859f-6a4313f0345f","order_by":1,"name":"Kazuki Sakurai","email":"","orcid":"","institution":"University of Warsaw","correspondingAuthor":false,"prefix":"","firstName":"Kazuki","middleName":"","lastName":"Sakurai","suffix":""}],"badges":[],"createdAt":"2025-10-05 11:08:12","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-7784331/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-7784331/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":100737143,"identity":"aa9c6872-5e9b-4820-8553-821b0c54dc2e","added_by":"auto","created_at":"2026-01-20 22:49:29","extension":"json","order_by":0,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":5311,"visible":true,"origin":"","legend":"","description":"","filename":"3597c833df78499fb21a584b8a5c7a79.json","url":"https://assets-eu.researchsquare.com/files/rs-7784331/v1/b0424d97b7b4c2130eb3020e.json"},{"id":100737295,"identity":"4b30232b-140d-407a-a6ee-9d7759ac6314","added_by":"auto","created_at":"2026-01-20 22:52:35","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":1961846,"visible":true,"origin":"","legend":"","description":"","filename":"QIPsubmission4.pdf","url":"https://assets-eu.researchsquare.com/files/rs-7784331/v1_covered_7095811d-782b-4175-821b-13cab2a9b17b.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Extremal Magic States from Symmetric Lattices","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-information-processing","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"qinp","sideBox":"Learn more about [Quantum Information Processing](http://link.springer.com/journal/11128)","snPcode":"11128","submissionUrl":"https://submission.nature.com/new-submission/11128/3","title":"Quantum Information Processing","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"stoa","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false},"keywords":"","lastPublishedDoi":"10.21203/rs.3.rs-7784331/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-7784331/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"Magic, a key quantum resource beyond entanglement, remains poorly understood in terms of its structure and classification. In this paper, we demonstrate a striking connection between high-dimensional symmetric lattices and quantum magic states. By mapping vectors from the $E_8, B W_{16}$, and $E_6$ lattices into Hilbert space, we construct and classify stabiliser and maximal magic states for two-qubit, three-qubit and one-qutrit systems. In particular, this geometric approach allows us to construct, for the first time, closed-form expressions for the maximal magic states in the three-qubit and one-qutrit systems, and to conjecture their total counts. In the three-qubit case, we further classify the extremal magic states according to their entanglement structure. We also examine the distinctive behaviour of one-qutrit maximal magic states with respect to Clifford orbits. Our findings suggest that deep algebraic and geometric symmetries underlie the structure of extremal magic states.","manuscriptTitle":"Extremal Magic States from Symmetric Lattices","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2026-01-20 22:36:32","doi":"10.21203/rs.3.rs-7784331/v1","editorialEvents":[{"type":"communityComments","content":0},{"type":"reviewersInvited","content":"","date":"2026-01-16T17:19:43+00:00","index":"","fulltext":""},{"type":"editorAssigned","content":"","date":"2025-10-10T08:43:19+00:00","index":"","fulltext":""},{"type":"checksComplete","content":"","date":"2025-10-10T08:40:24+00:00","index":"","fulltext":""},{"type":"submitted","content":"Quantum Information Processing","date":"2025-10-05T10:54:40+00:00","index":"","fulltext":""}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"quantum-information-processing","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"qinp","sideBox":"Learn more about [Quantum Information Processing](http://link.springer.com/journal/11128)","snPcode":"11128","submissionUrl":"https://submission.nature.com/new-submission/11128/3","title":"Quantum Information Processing","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"stoa","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false}}],"origin":"","ownerIdentity":"1f227e15-af33-4fdc-a7d7-3e353fa2187b","owner":[],"postedDate":"January 20th, 2026","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"under-review","subjectAreas":[],"tags":[],"updatedAt":"2026-01-20T22:36:32+00:00","versionOfRecord":[],"versionCreatedAt":"2026-01-20 22:36:32","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-7784331","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-7784331","identity":"rs-7784331","version":["v1"]},"buildId":"XKTyCvWXoU3ODBz1xrDgd","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}