[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-83023-en":3,"doc-seo-83023-105":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},83023,7971461740909,"Levi","https://ap-avatar.wpscdn.com/davatar_155a257f0dc6eb9ab79c44ca47cae57d",8,"Research & Report","QUBO Modeling of Module Learning With Errors Stability and Scaling in Post-Quantum Cryptography","Lattice-based post-quantum cryptography depends on the hardness of Learning With Errors (LWE) and Module Learning With Errors (MLWE). The work presents a constructive method to encode small MLWE instances as Quadratic Unconstrained Binary Optimization (QUBO) models for quantum annealing. Secret coefficients and explicit error variables are represented in one unified binary optimization, enabling joint recovery via the ground state. The paper analyzes stability under additive perturbations, characterizes an admissible noise region as a convex polytope, and validates scaling and robustness with exact simulations.","arXiv :2607 .05973v1 [ quant-ph] 7 Jul 2026  \nQUBO Modeling of Module Learning With Errors: Stability and Scaling in Post-Quantum Cryptography  \nRuturaj Khamitkar 1 , Durga Pritam Suggisetti2 ,3 ,  \nSoujanya Chatti3 , Varsha Sambhaje4 ∗,  \nand Durga Dasari5∗  \n1 D Y Patil International University, Akurdi, Maharashtra 411044, India  \n2 Birla Institute of Technology and Science Pilani, Dubai, UAE  \n3 QbitForce Quantum Pvt. Ltd. , Vijayawada, AP 520012, India  \n4 Department of Computer Science and Engineering, SRM University-AP, Amaravati 522240, India  \n53. Physikalisches Institut, University of Stuttgart, Stuttgart 70569, Germany  \nAbstract  \nLattice-based post-quantum cryptography relies on the hardness of the Learning With Errors (LWE) and Module Learning With Errors (MLWE) problems. This work introduces a constructive framework for encoding small MLWE instances as Quadratic Unconstrained Binary Optimization (QUBO) models suitable for quantum annealing. The formulation jointly represents secret coefficients and explicit error variables within a unified binary optimization structure, enabling their simultaneous recovery from the ground-state solution. Beyond the encoding, we develop a stability analysis of the resulting optimization landscape under additive perturbations. We show that the admissible noise region forms a convex polytope defined by competing candidate secrets, and establish an equivalent characterization in terms of the QUBO energy gap between the optimal and second-best solutions. Numerical experiments on low-dimensional benchmark instances using exact simulation demonstrate correct recovery of both secret and discretized error vectors, and confirm consistency between geometric stability regions and energy-gap behavior. We further quantify the scaling of logical variables and embedding overhead with increasing MLWE dimensions to assess feasibility on quantum annealing architectures. The results establish a systematic connection between MLWE problems and quantum optimization while providing a framework for analyzing robustness properties of QUBO formulations. Although current quantum annealing hardware remains insufficient for cryptographically relevant parameters, the proposed methodology offers a structured basis for studying lattice-based problems in quantum optimization settings without implying a practical threat to standardized post-quantum schemes.  \nKeywords: quantum annealing, QUBO, Module Learning With Errors, lattice-based cryptography, postquantum cryptography.  \n1 Introduction  \nQuantum annealing (QA) has emerged as an important optimization paradigm for problems that can be encoded as energy minimization over binary variables. Rather than executing a circuit model of computation, QA maps a cost function to an effective Hamiltonian and seeks low-energy configurations through an annealing process [1, 2] . This perspective has made QA attractive in application domains such as scheduling, routing, materials design, and structured combinatorial search [3, 4, 5] . At the same time, the ability to express a problem as a quadratic binary objective, often in QUBO form, has become increasingly valuable beyond physics, because it provides a common language shared by quantum annealers, Ising machines, and a variety of classical heuristics.  \nIn parallel, post-quantum cryptography (PQC) has become a central research area because widely deployed public-key systems such as RSA and elliptic-curve cryptography are vulnerable to Shor’s algorithm in a sufficiently powerful quantum setting [6] . Among the main PQC families, lattice-based constructions are particularly  \n∗ Corresponding authors  \nprominent because they combine strong worst-case-to-average-case hardness evidence with efficient implementations. LWE, introduced by Regev, is one of the foundational assumptions in this area [7, 8] . In its basic form, one is given noisy linear relations of the form  \nbi = ⟨ai , s⟩ + ei (mod q), (1)  \nwhere s is the hidden s","cbCaiuR1zXN0jQOl","https://ap.wps.com/l/cbCaiuR1zXN0jQOl","pdf",1549398,3,1,14,"English","en",105,"# Introduction\n## Background: Quantum annealing as optimization\n## Post-quantum cryptography and lattice assumptions\n## Motivation: QUBO/Ising encodings for lattice problems\n## Gap: joint secret-error modeling and stability analysis\n# Method and Contributions\n## Constructive MLWE-to-QUBO encoding\n## Joint recovery from ground states\n## Stability analysis under perturbations\n## Numerical experiments and exact simulation\n## Scaling and embedding overhead on annealing architectures","[{\"question\":\"What does the paper propose for MLWE problems in quantum annealing settings?\",\"answer\":\"It introduces a constructive framework that encodes small MLWE instances as QUBO models suitable for quantum annealing, using a unified binary optimization representation.\"},{\"question\":\"How are secret coefficients and error variables handled in the proposed QUBO formulation?\",\"answer\":\"The formulation jointly represents secret coefficients and explicit error variables inside the same QUBO structure, allowing simultaneous recovery from the ground-state solution.\"},{\"question\":\"How does the paper assess robustness of the optimization landscape?\",\"answer\":\"It develops a stability analysis under additive perturbations, characterizing the admissible noise region as a convex polytope and relating it to the QUBO energy gap between the best and second-best solutions.\"}]",1784184723,35,{"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},"qubo-modeling-of-module-learning-with-errors-stability-and-scaling-in-post-quantum-cryptography","",{"@graph":36,"@context":85},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,50],{"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":20},"https://docshare.wps.com/document/research-report/",{"item":51,"name":13,"@type":43,"position":52},"https://docshare.wps.com/document/qubo-modeling-of-module-learning-with-errors-stability-and-scaling-in-post-quantum-cryptography/83023/",4,{"url":51,"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","2026-07-24","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},"What does the paper propose for MLWE problems in quantum annealing settings?","Question",{"text":75,"@type":76},"It introduces a constructive framework that encodes small MLWE instances as QUBO models suitable for quantum annealing, using a unified binary optimization representation.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How are secret coefficients and error variables handled in the proposed QUBO formulation?",{"text":80,"@type":76},"The formulation jointly represents secret coefficients and explicit error variables inside the same QUBO structure, allowing simultaneous recovery from the ground-state solution.",{"name":82,"@type":73,"acceptedAnswer":83},"How does the paper assess robustness of the optimization landscape?",{"text":84,"@type":76},"It develops a stability analysis under additive perturbations, characterizing the admissible noise region as a convex polytope and relating it to the QUBO energy gap between the best and second-best 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