[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-83468-en":3,"doc-seo-83468-105":29,"detail-sidebar-cat-0-en-105":90},{"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":20,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":13,"seo_description":14,"update_tm":27,"read_time":28},83468,1099513958762,"Logic","https://ap-avatar.wpscdn.com/avatar/1000023916a998db790?x-image-process=image/resize,m_fixed,w_180,h_180&k=1784791008015729253",8,"Research & Report","Generative Modeling of Quantum Distribution with Functional Flow Matching","Powerful diffusion and flow-matching generative models have made it possible to model complex distributions, yet learning quantum distributions remains difficult because quantum states must preserve physically meaningful properties. Quantum Flow Matching (QFM) is presented as a generative model that learns quantum distributions using spin Wigner functions together with functional flow matching in function space. By mapping density matrices into spin Wigner functions and reconstructing new states, QFM produces multiqubit quantum distributions that match physical quantities including trace, purity, and entanglement entropy.","Generative Modeling of Quantum Distribution with Functional Flow Matching  \nJaehoon Hahm * 1 2 Tak Hur * 1 Joonseok Lee 2 3 Daniel K. Park 1 4  \narXiv :2607 .0030 1v 1 [ cs .LG] 1 Jul 2026  \nAbstract  \nThe emergence of powerful deep generative models based on diffusion and flow matching has enabled the learning and modeling of complex distributions. Learning quantum distributions, however, remains challenging due to the inherent difficulty of accurately modeling the meaningful physical properties of quantum states. We propose Quantum Flow Matching (QFM), a novel generative model designed to learn quantum distribution by utilizing spin Wigner function and flow matching.  \nBy converting density matrix into the spin Wigner function and leveraging functional flow matching to learn distributions in function space, QFM enables accurate and effective learning of multiqubit quantum distributions. We demonstrate the effectiveness of our method by evaluating physical quantities such as trace, purity, and entanglement entropy of the generated quantum states, accurately capturing the underlying physics of the given quantum distributions.  \n1. Introduction  \nDespite the unprecedented success of deep generative models, such as diffusion models (Song et al., 2020 ; Ho et al., 2020) and flow matching (Lipman et al., 2022), learning the distributions of quantum states remains a challenging task. Leveraging state-of-the-art machine learning methods to model quantum states has been an interesting research direction (Carrasquilla et al., 2019 ; Carleo et al., 2019) . However, applying modern generative models directly to learn quantum distributions has not been successful for several reasons. First, existing diffusion and flow matching methods are solely focused on learning representations of  \n*Equal contribution 1Department of Statistics and Data Science, Yonsei University, Seoul, Korea 2 Graduate School of Data Science, Seoul National University, Seoul, Korea 3 Google Research, Mountain View, California, United States 4Department of Applied Statistics, Yonsei University, Seoul, Korea. Correspondence to: Daniel K. Park \u003C[dkd.park@yonsei.ac.kr](dkd.park@yonsei.ac.kr) >.  \nAccepted as an extended abstract at QTML 2024 .  \n{ρi }1 ∼ 􀀢  \nW(Ω) = Tr[ρU(Ω)ΠU†(Ω)]  \nConvert to spin Wigner function  \n| ρgen |  |  |\n| --- | --- | --- |\n|  | +i |  |\n\nρ = dim(ρ)∫Ω Wρ(Ω)U(Ω)ΠU†(Ω)dΩ Reconstruction  \nW0 W1  \nWT = ϕT(W0) = ∫0T ut(ϕt(W0))dt  \nQuantum Flow Matching (QFM)  \nFigure 1 . Overall procedure of QFM. We bypass the direct learning of quantum states by converting them into spin Wigner function and learn the underlying distribution. This approach allows us to effectively generate physically valid and accurate quantum states, which was not achievable by direct learning from density matrices.  \nclassical data such as images or classical PDEs. These are inadequate for applications requiring consistency with important physical quantities such as purity, entanglement entropy, and quantum phases of matter. Second, complexvalued density matrices pose a significant challenge due to the sign structure problem, making previous methods unsuitable for handling them (Westerhout et al., 2020 ; Dugan et al., 2023) .  \nTo address this challenge, we propose Quantum Flow Matching (QFM), a novel generative modeling method that leverages flow matching to effectively learn the distributions of quantum states. The procedure of QFM, summarized in Figure 1, can be illustrated in the following steps. First, we convert quantum states into informationally complete functional representations using spin Wigner functions. Second, we employ Functional Flow Matching (FFM) to learn the distribution of spin Wigner functions in function space. This allows us to generate new spin Wigner functions that accurately reflect the underlying quantum distribution. Finally, the new quantum states are reconstructed from these generated spin Wigner functions.  \nOur contributions are summarized as follo","cbCaivXIwPimtrIk","https://ap.wps.com/l/cbCaivXIwPimtrIk","pdf",1580401,4,1,"English","en",105,"# Introduction\n# Learning Quantum Distribution with Flow Matching\n## Quantum State to Spin Wigner Function\n## Quantum Flow Matching (QFM)","[{\"question\":\"What problem does Quantum Flow Matching (QFM) address?\",\"answer\":\"QFM targets the difficulty of learning quantum-state distributions while accurately preserving key physical properties of quantum systems.\"},{\"question\":\"How does QFM represent quantum states for learning?\",\"answer\":\"QFM converts a density matrix into a spin Wigner function, turning quantum information into an informationally complete functional representation.\"},{\"question\":\"How is the quality of generated quantum states evaluated?\",\"answer\":\"The method is assessed by computing physical quantities such as trace, purity, and entanglement entropy from the generated quantum states.\"}]",1784188175,10,{"code":4,"msg":30,"data":31},"ok",{"site_id":24,"language":23,"slug":32,"title":13,"keywords":33,"description":14,"schema_data":34,"social_meta":85,"head_meta":87,"extra_data":89,"updated_unix":27},"generative-modeling-of-quantum-distribution-with-functional-flow-matching","",{"@graph":35,"@context":84},[36,52,67],{"@type":37,"itemListElement":38},"BreadcrumbList",[39,43,47,50],{"item":40,"name":41,"@type":42,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":44,"name":45,"@type":42,"position":46},"https://docshare.wps.com/document/","Document",2,{"item":48,"name":12,"@type":42,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":42,"position":20},"https://docshare.wps.com/document/generative-modeling-of-quantum-distribution-with-functional-flow-matching/83468/",{"url":51,"name":13,"@type":53,"author":54,"headline":13,"publisher":56,"fileFormat":59,"inLanguage":23,"description":14,"dateModified":60,"datePublished":61,"encodingFormat":59,"isAccessibleForFree":62,"interactionStatistic":63},"DigitalDocument",{"name":9,"@type":55},"Person",{"url":40,"name":57,"@type":58},"DocShare","Organization","application/pdf","2026-07-25","2026-07-16",true,{"@type":64,"interactionType":65,"userInteractionCount":20},"InteractionCounter",{"@type":66},"ViewAction",{"@type":68,"mainEntity":69},"FAQPage",[70,76,80],{"name":71,"@type":72,"acceptedAnswer":73},"What problem does Quantum Flow Matching (QFM) address?","Question",{"text":74,"@type":75},"QFM targets the difficulty of learning quantum-state distributions while accurately preserving key physical properties of quantum systems.","Answer",{"name":77,"@type":72,"acceptedAnswer":78},"How does QFM represent quantum states for learning?",{"text":79,"@type":75},"QFM converts a density matrix into a spin Wigner function, turning quantum information into an informationally complete functional representation.",{"name":81,"@type":72,"acceptedAnswer":82},"How is the quality of generated quantum states evaluated?",{"text":83,"@type":75},"The method is assessed by computing physical quantities such as trace, purity, and entanglement entropy from the generated quantum states.","https://schema.org",{"og:url":51,"og:type":86,"og:title":13,"og:site_name":57,"og:description":14},"article",{"robots":88,"canonical":51},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":91},[92,96,100,104,109,114,119,122,127,130,133],{"id":21,"doc_module":4,"doc_module_name":45,"category_name":93,"show_sort_weight":94,"slug":95},"Story & Novel",90,"story-novel",{"id":46,"doc_module":4,"doc_module_name":45,"category_name":97,"show_sort_weight":98,"slug":99},"Literature",80,"literature",{"id":20,"doc_module":4,"doc_module_name":45,"category_name":101,"show_sort_weight":102,"slug":103},"Exam",70,"exam",{"id":105,"doc_module":4,"doc_module_name":45,"category_name":106,"show_sort_weight":107,"slug":108},5,"Comic",60,"comic",{"id":110,"doc_module":4,"doc_module_name":45,"category_name":111,"show_sort_weight":112,"slug":113},6,"Technology",50,"technology",{"id":115,"doc_module":4,"doc_module_name":45,"category_name":116,"show_sort_weight":117,"slug":118},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":45,"category_name":12,"show_sort_weight":120,"slug":121},30,"research-report",{"id":123,"doc_module":4,"doc_module_name":45,"category_name":124,"show_sort_weight":125,"slug":126},9,"Religion & Spirituality",20,"religion-spirituality",{"id":125,"doc_module":4,"doc_module_name":45,"category_name":128,"show_sort_weight":125,"slug":129},"World Cup","world-cup",{"id":28,"doc_module":4,"doc_module_name":45,"category_name":131,"show_sort_weight":28,"slug":132},"Lifestyle","lifestyle",{"id":134,"doc_module":4,"doc_module_name":45,"category_name":135,"show_sort_weight":105,"slug":136},19,"General","general"]