[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-81614-en":3,"doc-seo-81614-105":30,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":13,"seo_description":14,"update_tm":28,"read_time":29},81614,34359740700684,"Finn","https://ap-avatar.wpscdn.com/avatar/1f400023980c374ae676?_k=1777273430885731487",8,"Research & Report","Adaptive Encoding Strategy for Quantum Annealing in Mixed-Variable Engineering Optimization","Mixed discrete–continuous optimization is central to engineering design, especially when discrete choices couple with continuous fields such as displacement and stress. Quantum annealing offers global-search potential, but its binary nature makes encoding continuous quantities a key difficulty. Existing methods either separate coupled variables or rely on fixed bit-depth representations, both of which can reduce quality or scalability. This work introduces an adaptive encoding that updates representable ranges during optimization, enabling accurate coupled formulations while improving precision under fixed binary budgets.","arXiv :2603 . 17506v2 [ cs .CE] 10 Jul 2026  \nAdaptive Encoding Strategy for Quantum Annealing in Mixed-Variable Engineering Optimization  \nFabian Key  [fabian.key@tuwien.ac.at](fabian.key@tuwien.ac.at)[ ](fabian.key@tuwien.ac.at)TU Wien, Vienna, Austria  \nMayu Muramatsu  [muramatsu@mech.keio.ac.jp](muramatsu@mech.keio.ac.jp)  \nKeio University, Yokohama, Kanagawa, Japan  \nLukas Freinberger  [lukas.freinberger@tuwien.ac.at](lukas.freinberger@tuwien.ac.at)  \nTU Wien, Vienna, Austria  \nNorbert Hosters  [hosters@cats.rwth-aachen.de](hosters@cats.rwth-aachen.de)  \nRWTH Aachen University, Aachen, Germany  \nAbstract  \nMixed discrete–continuous optimization is central to engineering design, exemplified by structural design where discrete choices interact with continuous fields such as displacement and stress. These problems are difficult due to high-dimensional, complex search spaces. To tackle them, quantum annealing (QA) is promising, yet its native binary nature supports only discrete variables, making accurate and efficient encodings of continuous quantities a central challenge. Existing approaches either split the coupled problem, mapping discrete decisions to QA while solving continuous fields classically, or use fixed-bit-depth encodings. The former compromises QA’s global search advantages; the latter can underrepresent dynamic range or inflate the number of binary variables, harming solution quality and scalability. In line with this, we show that simply increasing bit depth can even degrade performance on current QA hardware, underscoring the need for alternative encodings. In response, we introduce an adaptive encoding strategy for continuous variables in QA that enables efficient treatment of coupled mixed-variable problems. To this end, we propose an update strategy for the representable ranges of the continuous variables. We demonstrate its utility by integrating it into the minimum complementary energy formulation for structural design optimization, which provides a single, coupled constrained problem over stress and design variables. We apply a quadratic penalty method where, at each iteration, we update the representation of the continuous variables while targeting the full original objective, preserving QA’s global search capability. On a published structural design benchmark, namely the size optimization of a composite rod, our adaptive encoding improves solution quality by orders of magnitude under a fixed binary variable budget, demonstrating a superior precision–resource trade-off. Since the framework generalizes beyond structural design, it offers practical guidance for encoding continuous variables for QA and indicates that adaptive representations can enhance precision on current hardware.  \nKeywords  \nApplied quantum computing; quantum annealing; adaptive encoding strategy; mixed discrete–continuous optimization; structural design optimization; composite rod; computational mechanics  \nData availability  \nThe data presented in this study are available at: [https://doi.org/10.48436/vmpfx-80w27](https://doi.org/10.48436/vmpfx-80w27) .  \nFunding  \nThis research was funded in whole or in part by the Austrian Science Fund (FWF) 10.55776/ESP2444325 . For open access purposes, the author has applied a CC BY public copyright license to any author accepted manuscript version arising from this submission.  \nAcknowledgments  \nThe authors gratefully acknowledge the J¨ulich Supercomputing Centre ([https://www.fz-juelich.de/ias/](https://www.fz-juelich.de/ias/)[ ](https://www.fz-juelich.de/ias/)[jsc](jsc)) [for funding this project by providing computing time on the D-Wave Advantage](for funding this project by providing computing time on the D-Wave Advantage)™ System JUPSI through the J¨ulich UNified Infrastructure for Quantum computing (JUNIQ) .  \nThe authors acknowledge TU Wien Bibliothek for financial support through its Open Access Funding Program.  \nConflict of interest  \nThe authors declare no conflicts of interest. 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The encoding choice directly affects the number of binary variables, numerical precision, and solution quality.\"},{\"question\":\"What limitations do existing encoding approaches face?\",\"answer\":\"Splitting the coupled problem can compromise quantum annealing’s global search advantages, while fixed-bit-depth encodings may fail to represent the dynamic range well or inflate the binary variable count, harming quality and scalability.\"},{\"question\":\"How does the proposed adaptive encoding improve performance on current QA hardware?\",\"answer\":\"The strategy updates the representable ranges of continuous variables at each iteration using an update rule, while targeting the full original objective through a coupled constrained formulation. On a structural benchmark, it improves solution quality by orders of magnitude under a fixed binary variable budget.\"}]",1784174790,68,{"code":4,"msg":31,"data":32},"ok",{"site_id":25,"language":24,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":28},"adaptive-encoding-strategy-for-quantum-annealing-in-mixed-variable-engineering-optimization","",{"@graph":36,"@context":85},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,51],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":47},"https://docshare.wps.com/document/","Document",2,{"item":49,"name":12,"@type":43,"position":50},"https://docshare.wps.com/document/research-report/",3,{"item":52,"name":13,"@type":43,"position":20},"https://docshare.wps.com/document/adaptive-encoding-strategy-for-quantum-annealing-in-mixed-variable-engineering-optimization/81614/",{"url":52,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":24,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":41,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-25","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"Why is encoding continuous variables challenging for quantum annealing in mixed-variable engineering optimization?","Question",{"text":75,"@type":76},"Quantum annealing natively supports only binary variables, so continuous fields must be encoded explicitly. The encoding choice directly affects the number of binary variables, numerical precision, and solution quality.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"What limitations do existing encoding approaches face?",{"text":80,"@type":76},"Splitting the coupled problem can compromise quantum annealing’s global search advantages, while fixed-bit-depth encodings may fail to represent the dynamic range well or inflate the binary variable count, harming quality and scalability.",{"name":82,"@type":73,"acceptedAnswer":83},"How does the proposed adaptive encoding improve performance on current QA hardware?",{"text":84,"@type":76},"The strategy updates the representable ranges of continuous variables at each iteration using an update rule, while targeting the full original objective through a coupled constrained formulation. 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