[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-117780-en":3,"doc-seo-117780-105":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},117780,687197100911,"Himbo","https://ap-avatar.wpscdn.com/avatar/a000239b6f1da00475?x-image-process=image/resize,m_fixed,w_180,h_180&k=1785132997149421697",8,"Research & Report","Expressivity of Variational Quantum Machine Learning on the Boolean Cube","Categorical data is central to machine learning, often mapped to real-valued functions on the n-dimensional Boolean cube. Variational quantum machine learning models embed classical inputs into parameterized quantum circuits, but commonly used schemes focus on continuous encodings. This work analyzes embeddings tailored for Boolean-valued data, deriving representability conditions using a phase embedding and a quantum random access code embedding. The results characterize which Boolean-cube functions can be represented by variational linear quantum models and ensembles, discuss benefits of each embedding and serial repetitions, and validate performance via simulations and experiments on IBM quantum processors with Qiskit Machine Learning.","Date of publication xxxx 00, 0000, date of current version xxxx 00, 0000 .  \nDigital Object Identifier 10.1109/TQE.2020.DOI  \nExpressivity of Variational Quantum Machine Learning on the Boolean Cube  \nApr 2023  \nDYLAN HERMAN1 , RUDY RAYMOND2,3 , MUYUAN LI4 , NICOLAS ROBLES4 , ANTONIO MEZZACAPO4 , MARCO PISTOIA1  \n1Global Technology Applied Research, JPMorgan Chase, New York, New York 10017, USA  \n2IBM Quantum, IBM Research-Tokyo, 19-21 Nihonbashi Hakozaki-cho, Chuo-ku, Tokyo, 103-8510, Japan 3Quantum Computing Center, Keio University, 3-14-1 Hiyoshi, Kohoku-ku, Yokohama, Kanagawa, 223-8522, Japan 4IBM Quantum, IBM T. J. Watson Research Center, Yorktown Heights, New York 10598, USA  \nCorresponding author: Dylan Herman (email: [dylan.a.herman@jpmorgan.com](dylan.a.herman@jpmorgan.com)).  \n[ quant-ph] 21  \nABSTRACT Categorical data plays an important part in machine learning research and appears in a variety of applications. Models that can express large classes of real-valued functions on the Boolean cube are useful for problems involving discrete-valued data types, including those which are not Boolean. To this date, the commonly used schemes for embedding classical data into variational quantum machine learning models encode continuous values. Here we investigate quantum embeddings for encoding Boolean-valued data into parameterized quantum circuits used for machine learning tasks. We narrow down representability conditions for functions on the n-dimensional Boolean cube with respect to previously known results, using two quantum embeddings: a phase embedding and an embedding based on quantum random access codes. We show that for any real-valued function on the n-dimensional Boolean cube, there exists avariational linear quantum model based on a phase embedding using n qubits that can represent it and  \narXiv :2204 .05286v3  \nan ensemble of such models using d \u003C n qubits that can express any function with degree at most d. Additionally, we prove that variational linear quantum models that use the quantum random access code embedding can express functions on the Boolean cube with degree d 􀀔 d~~n~~3e using d ~~n~~3e qubits, and that an ensemble of such models can represent any function on the Boolean cube with degree d 􀀔 d ~~n~~3e. Furthermore, we discuss the potential benefits of each embedding and the impact of serial repetitions. Lastly, we demonstrate the use of the embeddings presented by performing numerical simulations and experiments on IBM quantum processors using the Qiskit machine learning framework.  \n INDEX TERMS quantum machine learning, variational quantum algorithms, expressivity, Fourier analysis, Boolean cube  \nI. INTRODUCTION  \nMachine learning problems involving categorical data are prevalent across many domains. The range of a categorical variable lies in a finite set, and each element in this set can be associated with an integer. Thus one can map a single categorical variable to multiple binary variables. If our goal is to perform supervised learning, then this converts the problem into learning a real-valued function on the n-dimensional Boolean (hyper)cube Bn = f0; 1gn. This implies that models that can express large classes of functions on the Boolean cube are useful for problems involving categorical data.In this article, we consider using variational quantum machine learning (VQML) [1] to fit real-valued functions of multiple binary variables. Thus, such models can be applied to regression or classifica-  \ntion tasks. Beyond machine learning, variational quantum algorithms [2] have been applied to chemistry [3]–[5], combinatorial optimization [6], [7], quantum linear systems [8], [9], and the simulation of quantum dynamics [10],[11] . When applied to supervised learning, VQML consists of using parameterized quantum circuits (PQCs) built from two types of circuit blocks: embedding blocks, which encode the inputs into a quantum system, and trainable blocks, where learnable parameters are adjusted in order to","cbCaitQz1jyubYAa","https://ap.wps.com/l/cbCaitQz1jyubYAa","pdf",1064226,1,18,"English","en",105,"# Abstract\n# Introduction\n## Categorical data and the Boolean cube formulation\n## VQML as parameterized quantum circuits with embedding and trainable blocks\n## Expressivity, Fourier-series connections, and data re-uploading\n# Embedding schemes and representability results\n## Phase embedding\n## Quantum random access code embedding\n# Applications and experiments","[{\"question\":\"What input encoding methods are studied for variational quantum machine learning on Boolean data?\",\"answer\":\"The work compares a phase embedding with an embedding based on quantum random access codes, focusing on encoding Boolean-valued inputs into parameterized quantum circuits.\"},{\"question\":\"What kind of representability guarantees are derived in the paper?\",\"answer\":\"The paper narrows down conditions under which functions on the n-dimensional Boolean cube can be represented, providing results for variational linear quantum models and ensembles based on function degree.\"},{\"question\":\"How are the theoretical findings validated?\",\"answer\":\"The study performs numerical simulations and experiments on IBM quantum processors, using the Qiskit machine learning framework to demonstrate the embeddings’ practical behavior.\"}]","Expressivity of Variational Quantum Machine Learning on the Boolean Cube | PDF",1785679508,45,{"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},"expressivity-of-variational-quantum-machine-learning-on-the-boolean-cube","",{"@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/expressivity-of-variational-quantum-machine-learning-on-the-boolean-cube/117780/",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},"What input encoding methods are studied for variational quantum machine learning on Boolean data?","Question",{"text":75,"@type":76},"The work compares a phase embedding with an embedding based on quantum random access codes, focusing on encoding Boolean-valued inputs into parameterized quantum circuits.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"What kind of representability guarantees are derived in the paper?",{"text":80,"@type":76},"The paper narrows down conditions under which functions on the n-dimensional Boolean cube can be represented, providing results for variational linear quantum models and ensembles based on function degree.",{"name":82,"@type":73,"acceptedAnswer":83},"How are the theoretical findings validated?",{"text":84,"@type":76},"The study performs numerical simulations and experiments on IBM quantum processors, using the Qiskit machine learning framework to demonstrate the embeddings’ practical behavior.","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"]