[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-83261-en":3,"doc-seo-83261-105":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":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":13,"seo_description":14,"update_tm":28,"read_time":29},83261,13056703019662,"Evangeline","https://ap-avatar.wpscdn.com/avatar/be000253a8e92610077?_k=1778726343310543188",8,"Research & Report","Towards Minimax Estimation of High-Order Functionals by Quantum Arguments","A quantum-computing-based approach addresses minimax estimation of high-order functionals for both classical discrete distributions and quantum mixed states. For large α, two estimators target Fα(P)=Σ pi^α and Fα(ρ)=tr(ρα), linking them to Rényi and Tsallis entropies. Both achieve the minimax optimal L2 rate α n−1 over α≲n≲α3−o(1), with support size or dimension far exceeding sample size. Sample complexity improves to n≈α versus previous O(α2) bounds, using a unified quantum-primitives framework that runs in linear time.","arXiv :2607 .07540v1 [ quant-ph] 8 Jul 2026  \nTowards Minimax Estimation of High-Order Functionals  \nby Quantum Arguments  \nQisheng Wang∗  \nAbstract  \nWe propose a novel approach to the minimax estimation of high-order functionals from the perspective of quantum computing. Specifically, for any real number α ≫ 1, we present two estimators, one for the classical functional Fα (P) = Ppαi of a discrete distribution P and the other for the quantum functional Fα (ρ) = tr(ρα ) of a mixed state ρ . These functionals have close connections with the R´enyi entropy and the Tsallis entropy. We show that both estimators achieve the minimax optimal L2 rate αn−1 in the range α ≲ n ≲ α3−o(1), where the support size S of P or the dimension of ρ can be much larger than the number of samples n. As a result, both estimators achieve the optimal sample complexity n ≍ α, improving upon the prior best upper bounds O(α2 ) established by Jiao, Venkat, Han, and Weissman (IEEE Trans. Inf. Theory 2017) for classical functionals and Chen and Wang (COLT 2025) for quantum functionals. Our estimators are constructed under a unified framework using quantum primitives and run in linear time on a quantum computer. This work reveals an unexpected path from quantum computing to statistics, suggesting a conceptually new methodology for functional estimation. It adds to the growing list of quantum proofs for classical theorems.  \n∗ Qisheng Wang is with the School of Computer Science, Shanghai Jiao Tong University, Shanghai 200240, China (e-mail: [QishengWang1994@gmail.com](QishengWang1994@gmail.com)).  \nContents  \n1 Introduction 1  \n1.1 Main results ......................................... 3  \n1.2 Techniques ......................................... 4  \n1.2.1 Relate classical functionals to quantum functionals ............... 4  \n1.2.2 The construction of our estimator ........................ 5  \n1.2.3 Lower bounds .................................... 9  \n1.3 Related work ........................................ 10  \n1.4 Discussion .......................................... 10  \n2 Overview 10  \n2.1 Sample complexity ..................................... 12  \n2.2 Time efficiency ....................................... 12  \n3 Preliminaries 13  \n3.1 Basic notations ....................................... 13  \n3.2 Quantum computing .................................... 14  \n3.3 Hadamard test ....................................... 15  \n3.4 Quantum query algorithms with block-encoding ..................... 16  \n3.5 Quantum singular value transformation ......................... 16  \n3.6 Quantum samplizer ..................................... 17  \n4 Upper Bounds 18  \n4.1 Generalized SWAP test for block-encodings ....................... 18  \n4.2 Subroutines with queries .................................. 19  \n4.3 Samplization ........................................ 20  \n4.4 A baby estimator ...................................... 21  \n4.5 Final estimator ....................................... 25  \n5 Lower Bounds 27  \n5.1 Sample complexity lower bounds ............................. 27  \n5.2 MSE lower bounds ..................................... 30  \nReferences 31  \nA Proof of Lemma 3.3 39  \n1 Introduction  \nClassical functional estimation. Given n samples drawn from an unknown discrete probability distribution P = (p1 , p2 ,..., pS ) of alphabet size S, the estimation of the functionals of the distribution P of the form  \nS  \nF (P) =Xf(pi) (1)  \ni=1  \nhas been extensively investigated in the literature. This fundamental problem has strong applications in entropy estimation. For example, when f (x) = −x ln(x), the functional becomes the Shannon entropy H(P) = −Ppi ln(pi) [Sha48a, Sha48b], and its estimation has been thoroughly studied in a series of works [Pan03, BDKR05 , Pan04 , VV11a, VV11b, VV17 , JVHW15 , JVHW17 , WY16] .  \nIn particular, f (x) = xα with parameter α gives a family of information measures of the form  \nS  \nFα(P) := Xpαi . (2)  \ni=1  \nThis type of infor","cbCaif0h8oPX46t6","https://ap.wps.com/l/cbCaif0h8oPX46t6","pdf",596953,3,1,42,"English","en",105,"# Introduction\n## Main results\n## Techniques\n## Related work\n## Discussion\n# Overview\n## Sample complexity\n## Time efficiency\n# Preliminaries\n## Basic notations\n## Quantum computing\n## Hadamard test\n## Quantum query algorithms with block-encoding\n## Quantum singular value transformation\n## Quantum samplizer\n# Upper Bounds\n## Generalized SWAP test for block-encodings\n## Lower Bounds\n## Sample complexity lower bounds\n## MSE lower bounds","[{\"question\":\"What two types of functionals are estimated in this work?\",\"answer\":\"The paper estimates the classical high-order functional Fα(P)=Σ pi^α for a discrete distribution P and the quantum analogue Fα(ρ)=tr(ρα) for a mixed state ρ.\"},{\"question\":\"In what parameter range do the proposed estimators achieve the minimax optimal rate?\",\"answer\":\"They achieve the minimax optimal L2 rate α n−1 in the range α≲n≲α3−o(1).\"},{\"question\":\"How does the method improve upon previous sample complexity bounds?\",\"answer\":\"The estimators reduce sample complexity to n≈α, improving earlier best upper bounds of order O(α2) for both classical and quantum functionals.\"}]",1784186344,106,{"code":4,"msg":31,"data":32},"ok",{"site_id":25,"language":24,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":28},"towards-minimax-estimation-of-high-order-functionals-by-quantum-arguments","",{"@graph":36,"@context":85},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,50],{"item":41,"name":42,"@type":43,"position":21},"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":20},"https://docshare.wps.com/document/research-report/",{"item":51,"name":13,"@type":43,"position":52},"https://docshare.wps.com/document/towards-minimax-estimation-of-high-order-functionals-by-quantum-arguments/83261/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":24,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":41,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-23","2026-07-16",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 two types of functionals are estimated in this work?","Question",{"text":75,"@type":76},"The paper estimates the classical high-order functional Fα(P)=Σ pi^α for a discrete distribution P and the quantum analogue Fα(ρ)=tr(ρα) for a mixed state ρ.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"In what parameter range do the proposed estimators achieve the minimax optimal rate?",{"text":80,"@type":76},"They achieve the minimax optimal L2 rate α n−1 in the range α≲n≲α3−o(1).",{"name":82,"@type":73,"acceptedAnswer":83},"How does the method improve upon previous sample complexity bounds?",{"text":84,"@type":76},"The estimators reduce sample complexity to n≈α, improving earlier best upper bounds of order O(α2) for both classical and quantum 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