Self-contained relaxation-based dynamical Ising machines

Self-contained relaxation-based dynamical Ising 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 Self-contained relaxation-based dynamical Ising machines Mikhail Erementchouk, Aditya Shukla, Pinaki Mazumder This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-5469197/v1 This work is licensed under a CC BY 4.0 License Status: Under Review Version 1 posted 9 You are reading this latest preprint version Abstract Dynamical Ising machines are based on continuous dynamical systems evolving from a generic initial state to a state strongly related to the ground state of the classical Ising model on a graph. Reaching the ground state is equivalent to finding the maximum (weighted) cut of the graph, which presents the Ising machines as an alternative way to solving and investigating NP-complete problems. Among the dynamical models, relaxation-based models are distinguished by their relations with guarantees of performance achieved in time scaling polynomially with the problem size. However, the terminal states of such machines are essentially non-binary, necessitating special post-processing relying on disparate computing. We show that an Ising machine implementing a special continuous dynamical system (called the V2 model) solves the rounding problem dynamically. We prove that the V2 model, starting from an arbitrary non-binary state, terminates in a state that trivially rounds to a binary state with the cut at least as big as obtained by optimal rounding of the initial state. Besides showing that relaxation-based dynamical Ising machines can be made self-contained, this result presents a non-Boolean realization of solving a non-trivial information processing task on Ising machines. Moreover, we prove that if the initial state of the V2-machine is a random limited amplitude perturbation of a binary state, the machine progresses to a state with at least as high cut as that of the initial binary state. Since the probability of improving the cut is finite, this shows that the V2-machine with random agitations converges to a maximum cut state almost surely. dynamical computations combinatorial optimization Ising machines Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Under Review Version 1 posted Editorial decision: Revision requested 05 Jun, 2025 Reviews received at journal 05 Jun, 2025 Reviews received at journal 10 Mar, 2025 Reviewers agreed at journal 30 Dec, 2024 Reviewers agreed at journal 23 Dec, 2024 Reviewers invited by journal 16 Dec, 2024 Editor assigned by journal 16 Dec, 2024 Submission checks completed at journal 18 Nov, 2024 First submitted to journal 17 Nov, 2024 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-5469197","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":391061679,"identity":"8f242cb8-211d-4fec-abbc-8c5177246e45","order_by":0,"name":"Mikhail Erementchouk","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAx0lEQVRIiWNgGAWjYDADfhCRUECCDgnJBpAWA1K0GBwAUcRo0W3vffjoRsWdOuPzqxM/PDBgkOcXO4Bfi9mZ48bGOWeeSZjdeLtZAugww5mzEwhouZHGJp3bdhio5ewGkJYEg9uEtNx/xv47999hCeMZZzf/IE7LDTY25tyGwxIG/L3biLTlTBqzdM6xw5IzbvBus0gwkCDCL8ePMX7OqTnMz99/dvPNHxU28vzSBLQggARYpQSxykGA/wApqkfBKBgFo2AkAQDsLkU9swuOrgAAAABJRU5ErkJggg==","orcid":"","institution":"University of Michigan–Ann Arbor","correspondingAuthor":true,"prefix":"","firstName":"Mikhail","middleName":"","lastName":"Erementchouk","suffix":""},{"id":391061680,"identity":"2a91b74c-fd6b-49df-9445-279646748f83","order_by":1,"name":"Aditya Shukla","email":"","orcid":"","institution":"University of Michigan–Ann Arbor","correspondingAuthor":false,"prefix":"","firstName":"Aditya","middleName":"","lastName":"Shukla","suffix":""},{"id":391061681,"identity":"419ca9c1-1990-42e3-b815-2286648bed05","order_by":2,"name":"Pinaki Mazumder","email":"","orcid":"","institution":"University of Michigan–Ann Arbor","correspondingAuthor":false,"prefix":"","firstName":"Pinaki","middleName":"","lastName":"Mazumder","suffix":""}],"badges":[],"createdAt":"2024-11-17 09:38:01","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-5469197/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-5469197/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":71719643,"identity":"02fa704b-5081-4039-8a39-7efaad4dcb71","added_by":"auto","created_at":"2024-12-18 04:51:55","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":1018027,"visible":true,"origin":"","legend":"","description":"","filename":"ErementchouketalSelfALDIM.pdf","url":"https://assets-eu.researchsquare.com/files/rs-5469197/v1_covered_9480ec62-69f0-499f-95ff-4df5a8960276.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Self-contained relaxation-based dynamical Ising machines","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":"natural-computing","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"naco","sideBox":"Learn more about [Natural Computing](http://link.springer.com/journal/11047)","snPcode":"11047","submissionUrl":"https://submission.nature.com/new-submission/11047/3","title":"Natural Computing","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"em","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false},"keywords":"dynamical computations, combinatorial optimization, Ising machines","lastPublishedDoi":"10.21203/rs.3.rs-5469197/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-5469197/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eDynamical Ising machines are based on continuous dynamical systems evolving from a generic initial state to a state strongly related to the ground state of the classical Ising model on a graph. 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Besides showing that relaxation-based dynamical Ising machines can be made self-contained, this result presents a non-Boolean realization of solving a non-trivial information processing task on Ising machines. Moreover, we prove that if the initial state of the V2-machine is a random limited amplitude perturbation of a binary state, the machine progresses to a state with at least as high cut as that of the initial binary state. 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