[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-126977-en":3,"doc-seo-126977-105":30,"detail-sidebar-cat-0-en-105":95},{"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},126977,687207024478,"Liam","https://ap-avatar.wpscdn.com/davatar_a8503ba1806abce46bf441b54a3ca4cd",8,"Research & Report","Regularization of Riemannian optimization - Application to process tomography and quantum machine learning","Regularized Riemannian gradient descent is studied for optimizing quantum channels on Riemannian manifolds. The work analyzes how adding different regularization terms to the cost function affects convergence and solution quality. Motivated by Lasso-style ideas, penalties are introduced to favor low-rank quantum channels, expressed with as few Kraus operators as possible. Applications include quantum process tomography and quantum machine learning classification, where regularization can simplify the learned channel without reducing accuracy and helps reveal the minimum channel rank supported by the data.","Regularization of Riemannian optimization: Application to process tomography and quantum machine learning  \nFelix Soest 1 , Konstantin Beyer2,1 , and Walter T. Strunz 1  \n1 Institute of Theoretical Physics, TUD Dresden University of Technology, 01062, Dresden, Germany 2 Department of Physics, Stevens Institute of Technology, Hoboken, New Jersey 07030, USA  \narXiv :2404 . 19659v1 [ quant-ph] 30 Apr 2024  \nGradient descent algorithms on Riemannian manifolds have been used recently for the optimization of quantum channels. In this contribution, we investigate the influence of various regularization terms added to the cost function of these gradient descent approaches. Motivated by Lasso regularization, we apply penalties for large ranks of the quantum channel, favoring solutions that can be represented by as few Kraus operators as possible. We apply the method to quantum process tomography and a quantum machine learning problem. Suitably regularized models show faster convergence of the optimization as well as better fidelities in the case of process tomography. Applied to quantum classification scenarios, the regularization terms can simplify the classifying quantum channel without degrading the accuracy of the classification, thereby revealing the minimum channel rank needed for the given input data.  \n1 Introduction  \nQuantum channels or completely positive and trace preserving (CPT) maps describe all physically valid transformations of an arbitrary input quantum state to a corresponding output quantum state. In this sense, quantum channels are the most general framework of how information can be processed. Quantum computers, for example, implement specifically tailored state transformations to make use of quantum effects for the efficient processing of classical data encoded  \nin the input state. Noise influences the overall channel in a way which is usually detrimental for useful information processing. Similar problems arise in quantum communication schemes, where the transferred message is directly affected by imperfections of the transmission channel. Currently, many research efforts focus on the question of how to cope with this noise and how to make applications more resistant against it [1– 6] . In order to do so, it is often instrumental to identify the influence of the noise, that is, to analyze how the channel which transforms the quantum can be characterized [7–9] . Such atask is usually called quantum channel or process tomography and aims at finding a numerical representation of a channel that reproduces the experimentally obtained data [10–12] . Various methods have been proposed, ranging from linear inversion and maximum-likelihood methods [13–17] over convex optimization [18] and projection techniques [19 , 20] to machine learning approaches [21, 22] . Experimentally, process tomography has been implemented, for example, in superconducting qubits [23–28], optical setups [29 , 30], trapped ions [31], and nuclear spins [32] .  \nRecently, a method based on Riemannian gradient descent has been proposed for quantum process tomography [33] . This approach makes use of the fact that Kraus representations of quantum channels form a Stiefel manifold for which efficient optimization techniques exist [34 , 35] . The parametrization in terms of Kraus operators leads to a valid quantum channel by construction, thereby circumventing problems arising in other tomography methods, where the resulting maps  \nhave to be projected to the closest valid quantum state transformation after the optimization [33] .  \nIn general, a full-rank representation of a quantum channel scales exponentially with system size. However, often quantum channels can be well approximated by low-rank channels, especially if the channel emerges from a quantum circuit with limited connectivity and depth. Methods based on matrix product state representations of the Choi matrix [22] or compression approaches [36–39] can be used in such a case, to enable tomo","cbCaiaA0YC1z7g35","https://ap.wps.com/l/cbCaiaA0YC1z7g35","pdf",827977,1,15,"English","en",105,"# Introduction\n## Quantum channels and process tomography\n## Riemannian gradient descent on the Stiefel manifold\n## Low-rank channels, Kraus non-uniqueness, and regularization motivation\n## Applications to quantum process tomography and quantum machine learning","[{\"question\":\"What problem does the paper address in Riemannian optimization of quantum channels?\",\"answer\":\"It studies how regularization terms change the behavior of Riemannian gradient descent when optimizing quantum channels, especially regarding convergence toward low-rank solutions.\"},{\"question\":\"How is the paper motivated by Lasso regularization?\",\"answer\":\"It introduces penalties that encourage solutions with a small effective rank, aiming for representations using as few Kraus operators as possible, analogous in spirit to sparsity-promoting Lasso regularization.\"},{\"question\":\"What are the regularization schemes analyzed in the paper?\",\"answer\":\"The paper evaluates three regularization terms: one based on the Hilbert-Schmidt norm of Kraus operators, one using the Choi-matrix purity, and an L1-norm regularization of the Stiefel vector representing the Kraus decomposition.\"},{\"question\":\"What benefits does regularization bring for process tomography and quantum classification?\",\"answer\":\"For process tomography it yields faster convergence and better fidelities, while for quantum classification it can simplify the learned quantum channel without degrading classification accuracy and can indicate the minimum channel rank required by the input data.\"}]","Regularization of Riemannian optimization - Application to process tomography and quantum machine learning | PDF",1785936001,38,{"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":90,"head_meta":92,"extra_data":94,"updated_unix":28},"regularization-of-riemannian-optimization-application-to-process-tomography-and-quantum-machine-learning","",{"@graph":36,"@context":89},[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/regularization-of-riemannian-optimization-application-to-process-tomography-and-quantum-machine-learning/126977/",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","2026-08-05",true,{"@type":65,"interactionType":66,"userInteractionCount":4},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81,85],{"name":72,"@type":73,"acceptedAnswer":74},"What problem does the paper address in Riemannian optimization of quantum channels?","Question",{"text":75,"@type":76},"It studies how regularization terms change the behavior of Riemannian gradient descent when optimizing quantum channels, especially regarding convergence toward low-rank solutions.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How is the paper motivated by Lasso regularization?",{"text":80,"@type":76},"It introduces penalties that encourage solutions with a small effective rank, aiming for representations using as few Kraus operators as possible, analogous in spirit to sparsity-promoting Lasso regularization.",{"name":82,"@type":73,"acceptedAnswer":83},"What are the regularization schemes analyzed in the paper?",{"text":84,"@type":76},"The paper evaluates three regularization terms: one based on the Hilbert-Schmidt norm of Kraus operators, one using the Choi-matrix purity, and an L1-norm regularization of the Stiefel vector representing the Kraus decomposition.",{"name":86,"@type":73,"acceptedAnswer":87},"What benefits does regularization bring for process tomography and quantum classification?",{"text":88,"@type":76},"For process tomography it yields faster convergence and better fidelities, while for quantum classification it can simplify the learned quantum channel without degrading classification accuracy and can indicate the minimum channel rank required by the input data.","https://schema.org",{"og:url":52,"og:type":91,"og:title":13,"og:site_name":59,"og:description":14},"article",{"robots":93,"canonical":52},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":96},[97,101,105,109,114,119,124,127,132,135,139],{"id":20,"doc_module":4,"doc_module_name":46,"category_name":98,"show_sort_weight":99,"slug":100},"Story & Novel",90,"story-novel",{"id":47,"doc_module":4,"doc_module_name":46,"category_name":102,"show_sort_weight":103,"slug":104},"Literature",80,"literature",{"id":53,"doc_module":4,"doc_module_name":46,"category_name":106,"show_sort_weight":107,"slug":108},"Exam",70,"exam",{"id":110,"doc_module":4,"doc_module_name":46,"category_name":111,"show_sort_weight":112,"slug":113},5,"Comic",60,"comic",{"id":115,"doc_module":4,"doc_module_name":46,"category_name":116,"show_sort_weight":117,"slug":118},6,"Technology",50,"technology",{"id":120,"doc_module":4,"doc_module_name":46,"category_name":121,"show_sort_weight":122,"slug":123},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":125,"slug":126},30,"research-report",{"id":128,"doc_module":4,"doc_module_name":46,"category_name":129,"show_sort_weight":130,"slug":131},9,"Religion & Spirituality",20,"religion-spirituality",{"id":130,"doc_module":4,"doc_module_name":46,"category_name":133,"show_sort_weight":130,"slug":134},"World Cup","world-cup",{"id":136,"doc_module":4,"doc_module_name":46,"category_name":137,"show_sort_weight":136,"slug":138},10,"Lifestyle","lifestyle",{"id":140,"doc_module":4,"doc_module_name":46,"category_name":141,"show_sort_weight":110,"slug":142},19,"General","general"]