[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-118775-en":3,"doc-seo-118775-105":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},118775,687197100911,"Himbo","https://ap-avatar.wpscdn.com/avatar/a000239b6f1da00475?x-image-process=image/resize,m_fixed,w_180,h_180&k=1785132997149421697",8,"Research & Report","On the Sample Complexity of Quantum Boltzmann Machine Learning","Quantum Boltzmann machines (QBMs) provide machine-learning models for classical and quantum data, and this work formalizes QBM learning via an operational definition based on the gap between target and model expectation values, accounting for the polynomial size of the dataset. Using relative entropy as a loss, the study avoids barren plateaus and proves that stochastic gradient descent can find a solution with at most polynomially many Gibbs state preparations. It further shows that parameter pre-training cannot reduce sample-complexity upper bounds and proposes strategies using mean-field, Gaussian Fermionic, and geometrically local Hamiltonians, validated numerically on quantum and classical datasets.","arXiv :2306 . 14969v1 [ quant-ph] 26 Jun 2023  \nOn the Sample Complexity of Quantum Boltzmann Machine Learning  \nLuuk Coopmans􀀃 and Marcello Benedettiy  \nQuantinuum, Partnership House, Carlisle Place, London SW1P 1BX, United Kingdom  \n(Dated: June 26, 2023)  \nQuantum Boltzmann machines (QBMs) are machine-learning models for both classical and quantum data. We give an operational de􀀌nition of QBM learning in terms of the di􀀋erence in expectation values between the model and target, taking into account the polynomial size of the data set. By using the relative entropy as a loss function this problem can be solved without encountering barren plateaus. We prove that a solution can be obtained with stochastic gradient descent using at most a polynomial number of Gibbs states. We also prove that pre-training on a subset of the QBM parameters can only lower the sample complexity bounds. In particular, we give pre-training strategies based on mean-􀀌eld, Gaussian Fermionic, and geometrically local Hamiltonians. We verify these models and our theoretical 􀀌ndings numerically on a quantum and a classical data set. Our results establish that QBMs are promising machine learning models trainable on future quantum devices.  \nINTRODUCTION  \nMachine learning (ML) research has developed into a mature discipline with applications that impact many di􀀋erent aspects of society. Neural network and deep learning architectures have been deployed for tasks such as facial recognition, recommendation systems, time series modeling, and for analyzing highly complex data in science. In addition, unsupervised learning and generative modeling techniques are widely used for text, image, and speech generation tasks, which many people encounter regularly via interaction with chatbots and virtual assistants. Thus, the development of new machine learning models and algorithms can have signi􀀌cant consequences for a wide range of industries, and more generally, society as a whole [1] .  \nRecently, researchers in quantum information science have asked the question of whether quantum algorithms can o􀀋er advantages over conventional machine learning algorithms. This has led to the development of quantum algorithms for gradient descent, classi􀀌cation, generative modeling, reinforcement learning, as well as many other tasks [2{6] . However, one cannot straightforwardly generalize results from the conventional ML realm to the quantum ML realm. One must carefully reconsider the data encoding, training complexity, and sampling in the quantum machine learning (QML) setting. For example, it is yet unclear how to e􀀎ciently embed large data sets into quantum states so that a genuine quantum speedup is achieved [7, 8] . Furthermore, as quantum states prepared on quantum devices can only be accessed via sampling, one cannot estimate properties with arbitrary precision. This gives rise to new problems, such as barren plateaus [9{14], that make the training of certain QML models challenging or even practically impossible.  \nIn this work, we show that a particular quantum generative model, the quantum Boltzmann machine [15{18](QBM) without hidden units, does not su􀀋er from these issues, and can be trained with polynomial sample complexity on future fault-tolerant quantum computers. QBMs are physics-inspired ML models that generalize the classical Boltzmann machines to quantum Hamiltonian ans􀁿atze. The Hamiltonian ansatz is de􀀌ned on a graph where each node represents a qubit and each vertex represents an interaction. The task is to learn the strengths of the interactions, such that samples from the quantum model mimic samples taken from the target data set. Quantum generative models of this kind could 􀀌nd use in ML for science problems, by learning approximate descriptions of the experimental data. QBMs could also play an important role as components of larger QML models [19{23] . This is similar to how classical BMs can provide good weight initializations for the training of deep ne","cbCaivYsJuxvMjIO","https://ap.wps.com/l/cbCaivYsJuxvMjIO","pdf",1093219,1,24,"English","en",105,"# Introduction\n## Motivation for quantum machine learning\n## Quantum Boltzmann machines and the training problem\n## Operational definition of QBM learning\n## Main results and numerical verification","[{\"question\":\"How is quantum Boltzmann machine learning defined in this work?\",\"answer\":\"QBM learning is defined operationally by requiring the difference in expectation values between the target data and the quantum model to be small, with precision scaling polynomially with the problem size.\"},{\"question\":\"What loss function is used to train the quantum Boltzmann machine?\",\"answer\":\"The training objective uses quantum relative entropy as the loss function between the target and the model expectations.\"},{\"question\":\"What training guarantee does the paper provide for QBMs?\",\"answer\":\"The paper proves that stochastic gradient descent can obtain a solution using at most a polynomial number of Gibbs state preparations, avoiding issues like barren plateaus.\"}]","On the Sample Complexity of Quantum Boltzmann Machine Learning | PDF",1785720179,60,{"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},"on-the-sample-complexity-of-quantum-boltzmann-machine-learning","",{"@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/on-the-sample-complexity-of-quantum-boltzmann-machine-learning/118775/",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","2026-08-03",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},"How is quantum Boltzmann machine learning defined in this work?","Question",{"text":75,"@type":76},"QBM learning is defined operationally by requiring the difference in expectation values between the target data and the quantum model to be small, with precision scaling polynomially with the problem size.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"What loss function is used to train the quantum Boltzmann machine?",{"text":80,"@type":76},"The training objective uses quantum relative entropy as the loss function between the target and the model expectations.",{"name":82,"@type":73,"acceptedAnswer":83},"What training guarantee does the paper provide for QBMs?",{"text":84,"@type":76},"The paper proves that stochastic gradient descent can obtain a solution using at most a polynomial number of Gibbs state preparations, avoiding issues like barren plateaus.","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},[93,97,101,105,109,114,119,122,127,130,134],{"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":29,"slug":108},5,"Comic","comic",{"id":110,"doc_module":4,"doc_module_name":46,"category_name":111,"show_sort_weight":112,"slug":113},6,"Technology",50,"technology",{"id":115,"doc_module":4,"doc_module_name":46,"category_name":116,"show_sort_weight":117,"slug":118},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":120,"slug":121},30,"research-report",{"id":123,"doc_module":4,"doc_module_name":46,"category_name":124,"show_sort_weight":125,"slug":126},9,"Religion & Spirituality",20,"religion-spirituality",{"id":125,"doc_module":4,"doc_module_name":46,"category_name":128,"show_sort_weight":125,"slug":129},"World Cup","world-cup",{"id":131,"doc_module":4,"doc_module_name":46,"category_name":132,"show_sort_weight":131,"slug":133},10,"Lifestyle","lifestyle",{"id":135,"doc_module":4,"doc_module_name":46,"category_name":136,"show_sort_weight":106,"slug":137},19,"General","general"]