[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-117144-en":3,"doc-seo-117144-105":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":4,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":27,"seo_description":14,"update_tm":28,"read_time":29},117144,549758252649,"Ivy","https://ap-avatar.wpscdn.com/avatar/8000253669c5317157?_k=1778319167496531819",8,"Research & Report","Quantum machine learning with indefinite causal order","Conventional quantum machine learning uses variational parameterized circuits with quantum gates applied in a fixed order, producing a restricted-Fourier-series form whose Fourier coefficients lack sufficient flexibility. This restriction can limit learning performance when variational parameters change. The work introduces indefinite causal order for quantum gates, enabling a superposition of different orders and thereby enhancing learning capability. Since current platforms often simulate only fixed-order learning structures, a revised simulation protocol implements indefinite causal order and verifies its positive impact on concrete learning tasks.","arXiv :2403 .03533v1 [ quant-ph] 6 Mar 2024  \nQuantum machine learning with indefinite causal order  \nNannan Ma, 1 P. Z. Zhao, 1 and Jiangbin Gong 1, 2, 3, ∗  \n1 Centre for Quantum Technologies, National University of Singapore, Singapore 117543, Singapore  \n2 Department of Physics, National University of Singapore, Singapore 117551, Singapore  \n3 Joint School of National University of Singapore and Tianjin University,  \nInternational Campus of Tianjin University, Binhai New City, Fuzhou 350207, China  \n(Dated: March 7, 2024)  \nIn a conventional circuit for quantum machine learning, the quantum gates used to encode the input parameters and the variational parameters are constructed with a fixed order. The resulting output function, which can be expressed in the form of a restricted Fourier series, has limited flexibility in the distributions of its Fourier coefficients. This indicates that a fixed order of quantum gates can limit the performance of quantum machine learning. Building on this key insight (also elaborated with examples), we introduce indefinite causal order to quantum machine learning. Because the indefinite causal order of quantum gates allows for the superposition of different orders, the performance of quantum machine learning can be significantly enhanced. Considering that the current accessible quantum platforms only allow to simulate a learning structure with a fixed order of quantum gates, we reform the existing simulation protocol to implement indefinite causal order and further demonstrate the positive impact of indefinite causal order on specific learning tasks. Our results offer useful insights into possible quantum effects in quantum machine learning.  \nI. INTRODUCTION  \nArising from intrinsic quantum features such as superposition and entanglement, quantum computation [1– 3] beating its classical counterparts is becoming a reality. Numerous quantum algorithms [4–7] and experimental demonstrations [8–11] have aimed to realize quantum advantages. Inspired by the capabilities of classicalcomputation-based machine learning [12] across various fields [13–15], the pursuit of a new era in machine learning founded on quantum computation [16–18] is natural and necessary to harness quantum advantages. In a typical quantum learning model, a variational parameterized quantum circuit [19–25], containing various quantum gates, serves a role analogous to the neural network in classical machine learning. The input vector x and variational parameters θ are encoded as gate parameters. The output y is represented by the expectation value of an observable measured on the final state. This learning structure gives rise to a mapping function fθ : x → y , which is designed to approximate the ground truth relation from x to y. The architecture of a quantum circuit fundamentally influences learning model’s capability, akin to the role of structure in classical neural networks. Given that different quantum gates are not commutable in general, the ordering of quantum gates plays an important role in a quantum circuit and consequently in a quantum learning model. Specifically, the learning capability of a quantum machine learning model can be analyzed in terms of the frequency spectrum and Fourier coefficients of a Fourier series that expresses the function fθ . The Fourier coefficients are determined by the multiplication of the quantum gates used in a learning  \n∗ [phygj@nus.edu.sg](phygj@nus.edu.sg)  \nmodel. Now if, as in conventional quantum machine learning, the quantum circuit has a fixed order of all involved quantum gates, then the Fourier coefficients of the associated learning model lacks certain flexibility in response to changes in the variational parameters and therefore limiting the learning ability of the quantum model. This key recognition motivates us to introduce indefinite causal order to relax this fixed-order limitationso as to achieve an enhanced learning ability. The proposal of using indefinite causal or","cbCaihBThoSYLAyS","https://ap.wps.com/l/cbCaihBThoSYLAyS","pdf",2081490,1,11,"English","en",105,"# Introduction\n## Fixed causal order constraint in quantum learning\n## Indefinite causal order as an enhanced learning strategy\n## Simulation of indefinite causal order on quantum circuits\n## Verification on specific learning tasks","[{\"question\":\"Why can a fixed gate order limit quantum machine learning performance?\",\"answer\":\"With a fixed ordering, the learning function can be expressed as a restricted Fourier series, and the resulting Fourier coefficients have limited flexibility. This reduces how effectively the model responds to changes in variational parameters.\"},{\"question\":\"How does indefinite causal order improve quantum machine learning?\",\"answer\":\"Indefinite causal order allows quantum gates to exist in a superposition of different orders. This added flexibility leads to more adaptable Fourier coefficients and improved learning ability.\"},{\"question\":\"How is indefinite causal order implemented when available platforms only simulate fixed-order circuits?\",\"answer\":\"The work reformulates the existing simulation protocol to implement indefinite causal order on quantum circuits, then demonstrates its benefits using specific learning-task examples.\"}]","Quantum machine learning with indefinite causal order | PDF",1785674095,28,{"code":4,"msg":31,"data":32},"ok",{"site_id":24,"language":23,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":28},"quantum-machine-learning-with-indefinite-causal-order","",{"@graph":36,"@context":85},[37,54,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,51],{"item":41,"name":42,"@type":43,"position":20},"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":53},"https://docshare.wps.com/document/quantum-machine-learning-with-indefinite-causal-order/117144/",4,{"url":52,"name":13,"@type":55,"author":56,"headline":13,"publisher":58,"fileFormat":61,"inLanguage":23,"description":14,"dateModified":62,"datePublished":62,"encodingFormat":61,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":57},"Person",{"url":41,"name":59,"@type":60},"DocShare","Organization","application/pdf","2026-08-02",true,{"@type":65,"interactionType":66,"userInteractionCount":4},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"Why can a fixed gate order limit quantum machine learning performance?","Question",{"text":75,"@type":76},"With a fixed ordering, the learning function can be expressed as a restricted Fourier series, and the resulting Fourier coefficients have limited flexibility. This reduces how effectively the model responds to changes in variational parameters.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does indefinite causal order improve quantum machine learning?",{"text":80,"@type":76},"Indefinite causal order allows quantum gates to exist in a superposition of different orders. This added flexibility leads to more adaptable Fourier coefficients and improved learning ability.",{"name":82,"@type":73,"acceptedAnswer":83},"How is indefinite causal order implemented when available platforms only simulate fixed-order circuits?",{"text":84,"@type":76},"The work reformulates the existing simulation protocol to implement indefinite causal order on quantum circuits, then demonstrates its benefits using specific learning-task examples.","https://schema.org",{"og:url":52,"og:type":87,"og:title":13,"og:site_name":59,"og:description":14},"article",{"robots":89,"canonical":52},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,120,123,128,131,135],{"id":20,"doc_module":4,"doc_module_name":46,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":47,"doc_module":4,"doc_module_name":46,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":53,"doc_module":4,"doc_module_name":46,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":46,"category_name":107,"show_sort_weight":108,"slug":109},5,"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":46,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":46,"category_name":117,"show_sort_weight":118,"slug":119},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":121,"slug":122},30,"research-report",{"id":124,"doc_module":4,"doc_module_name":46,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":46,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":132,"doc_module":4,"doc_module_name":46,"category_name":133,"show_sort_weight":132,"slug":134},10,"Lifestyle","lifestyle",{"id":136,"doc_module":4,"doc_module_name":46,"category_name":137,"show_sort_weight":106,"slug":138},19,"General","general"]