[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-123596-en":3,"doc-seo-123596-105":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":20,"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},123596,1099513958762,"Logic","https://ap-avatar.wpscdn.com/avatar/1000023916a998db790?x-image-process=image/resize,m_fixed,w_180,h_180&k=1784791008015729253",8,"Research & Report","Robust Dequantization of the Quantum Singular value Transformation and Quantum Machine Learning Algorithms","Several quantum algorithms for linear algebra and quantum machine learning have been dequantized in recent years, typically relying on classical access to the input via length-squared sampling. This work studies robustness under an approximate version of that access, defined through closeness in total variation distance to the ideal distribution. It shows how randomized linear algebra techniques can be adapted to this weaker setting and uses them to generalize recent low-rank and sparse-matrix dequantization frameworks based on Quantum Singular Value Transformation. Robust dequantization is then derived for multiple quantum machine learning tasks, including recommendation, supervised clustering, and low-rank matrix inversion.","arXiv :2304 .04932v1 [ quant-ph] 11 Apr 2023  \nRobust Dequantization of the Quantum Singular value Transformation and Quantum Machine Learning Algorithms  \nFranc¸ois Le Gall  \nNagoya University  \n[legall@math.nagoya-u.ac.jp](legall@math.nagoya-u.ac.jp)  \nAbstract  \nSeveral quantum algorithms for linear algebra problems, and in particular quantum machine learning problems, have been “dequantized” in the past few years. These dequantization results typically hold when classical algorithms can access the data via lengthsquared sampling. This assumption, which is standard in the 􀀂eld of randomized linear algebra, means that for a unit-norm vector u ∈ Cn, we can sample from the distribution pu : {1, . . . , n} → [0,1] de􀀂ned as pu(i) = |u(i)| 2 for each i ∈ {1, . . . , n} . Since this distribution corresponds to the distribution obtained by measuring the quantum state |ui in the computational basis, length-squared sampling access gives a reasonable classical analogue to the kind of quantum access considered in many quantum algorithms for linear algebra problems.  \nIn this work we investigate how robust these dequantization results are. We introduce the notion of approximate length-squared sampling, where classical algorithms are only able to sample from a distribution close to the ideal distribution in total variation distance. While quantum algorithms are natively robust against small perturbations, current techniques indequantization are not. Our main technical contribution is showing how many techniques from randomized linear algebra can be adapted to work under this weaker assumption as well. We then use these techniques to show that the recent low-rank dequantization framework by Chia, Gily´en, Li, Lin, Tang and Wang (JACM 2022) and the dequantization framework for sparse matrices by Gharibian and Le Gall (STOC 2022), which are both based on the Quantum Singular Value Transformation, can be generalized to the case of approximate length-squared sampling access to the input. We also apply these results to obtain a robust dequantization of many quantum machine learning algorithms, including quantum algorithms for recommendation systems, supervised clustering and low-rank matrix inversion.  \n1 Introduction  \n1.1 Background  \nThe quantum algorithm for matrix inversion by Harrow, Hassidim and Lloyd [22], often called simply the HHL algorithm, is a milestone in quantum algorithms for linear algebra. Given a well-conditioned invertible matrix A ∈ Cn ×n and a vector u ∈ Cn, the algorithm prepares in time poly (log n) a quantum state proportional to a vector ˆx ∈ Cn close to the solution of the system of equations Ax = u. This quantum state can then be used to compute some partial information about ˆx . The HHL algorithm has been very in􀀃uential for the development of quantum algorithms solving problems related to linear algebra, and has in particular lead to several quantum machine learning algorithms ([6, 28, 29, 31, 33, 34, 35, 38] for instance) .  \nAn important assumption in the HHL algorithm, and in most of these extensions as well, is that the input vector should be accessible as a quantum state: the quantum computer has access to a few copies of the quantum state |vi = ∑1 v (i)|ii on O(log n) qubits, for v = u/ kuk. In order for the HHL algorithm to be useful, it is thus crucial to be able to prepare this quantum state ef􀀂ciently. One concrete proposal is having the vector u stored in Quantum Random Access Memory (QRAM), in which case (when using the de􀀂nition of QRAM in [28]) one copy of |vi can be created in time poly(log n) . Similar assumptions are needed on how the quantum algorithm can access the matrix A. As discussed in [1], all these assumptions make dif􀀂cult to directly compare the performance of these quantum algorithms to the performance of classical methods.  \nA series of works initiated by Tang [36] has been investigating the importance of such assumptions for quantum machine learning. These results have shown that ","cbCaiiZsz7Tl6DsC","https://ap.wps.com/l/cbCaiiZsz7Tl6DsC","pdf",523475,1,55,"English","en",105,"# Introduction\n## Background","[{\"question\":\"What is length-squared sampling in the context of dequantization?\",\"answer\":\"It is the classical ability to sample indices according to pu(i)=|u(i)|^2 (normalized), which matches measurement outcomes of the quantum state |u⟩ in the computational basis.\"},{\"question\":\"How does the paper define approximate length-squared sampling?\",\"answer\":\"Approximate length-squared sampling allows sampling from a distribution that is close to the ideal length-squared distribution in total variation distance.\"},{\"question\":\"Which frameworks and quantum tasks does the paper generalize using approximate sampling?\",\"answer\":\"It generalizes low-rank dequantization and sparse-matrix dequantization frameworks based on the Quantum Singular Value Transformation, and applies them to robust dequantization of recommendation systems, supervised clustering, and low-rank matrix inversion.\"}]","Robust Dequantization of the Quantum Singular value Transformation and Quantum Machine Learning Algorithms | PDF",1785817554,139,{"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},"robust-dequantization-of-the-quantum-singular-value-transformation-and-quantum-machine-learning-algorithms","",{"@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/robust-dequantization-of-the-quantum-singular-value-transformation-and-quantum-machine-learning-algorithms/123596/",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","2026-08-04",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What is length-squared sampling in the context of dequantization?","Question",{"text":75,"@type":76},"It is the classical ability to sample indices according to pu(i)=|u(i)|^2 (normalized), which matches measurement outcomes of the quantum state |u⟩ in the computational basis.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does the paper define approximate length-squared sampling?",{"text":80,"@type":76},"Approximate length-squared sampling allows sampling from a distribution that is close to the ideal length-squared distribution in total variation distance.",{"name":82,"@type":73,"acceptedAnswer":83},"Which frameworks and quantum tasks does the paper generalize using approximate sampling?",{"text":84,"@type":76},"It generalizes low-rank dequantization and sparse-matrix dequantization frameworks based on the Quantum Singular Value Transformation, and applies them to robust dequantization of recommendation systems, supervised clustering, and low-rank matrix inversion.","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"]