[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-83757-en":3,"doc-seo-83757-105":30,"detail-sidebar-cat-0-en-105":96},{"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},83757,137441390410,"Hazel","https://ap-avatar.wpscdn.com/avatar/2000252f4ab5702993?_k=1776741390130283984",8,"Research & Report","On estimating operator norm distance, with optimal trace distance estimation when one state is pure","Investigates computational complexity of estimating the operator norm distance T∞(ρ0,ρ1)=∥ρ0−ρ1∥∞ (largest singular value) for n-qubit states given poly(n)-size state-preparation circuits. Presents efficient quantum estimators whose query complexity is rank-independent. When one state is pure, establishes an optimal Θ(1/ε) query estimator, giving poly(n) time for constant additive error and a rank-independent estimate for both T∞ and trace distance. Shows additional problems where hardness is BQP-complete and includes matching lower bounds.","arXiv :2607 .03905v1 [ quant-ph] 4 Jul 2026  \nOn estimating operator norm distance, with optimal trace distance estimation when one state is pure  \nYupan Liu∗1, Qisheng Wang†2, and Zhan Yu‡3  \n1 School of Computer and Communication Sciences, École Polytechnique Fédérale de Lausanne  \n2 School of Computer Science, Shanghai Jiao Tong University  \n3 Centre for Quantum Technologies, National University of Singapore  \nAbstract  \nWe investigate the computational complexity of estimating the operator norm distance T∞ (ρ0 , ρ 1 ) , defined via the operator norm ∥A∥∞ := σmax (A), where σmax (A) is the largest singular value of A, given poly(n)-size state-preparation circuits of n-qubit quantum states ρ0 and ρ 1 . We provide efficient quantum estimators for the operator norm distance whose complexity is independent of the rank (and thus the dimension) of the states:  \n(1) When one state is pure, we establish an optimal quantum estimator using Θ(1/ϵ) queries to the state-preparation circuits. Consequently, for constant additive error, say ϵ = 1/5, our estimator runs in poly(n) time. Since the operator norm distance T∞ (|ψ⟩⟨ψ|, ρ) is exactly half of the trace distance T(|ψ⟩⟨ψ|, ρ), our result gives a rankindependent query complexity for estimating T ∞ (|ψ⟩⟨ψ|, ρ) and T(|ψ⟩⟨ψ|, ρ), whereas the approaches due to van Apeldoorn, Cornelissen, Gilyén, and Nannicini (SODA 2023) and Wang and Zhang (TIT 2024) have query complexity scaling at least linearly with rank(ρ), which can be exp(n) in general. In addition, our query complexity matches the optimal bound when both states are pure by Wang (TIT 2024) .  \n(2) For gquerieesnertoatlhqeusantutate-mprsetapatreast,iownecialsrcouitpsr,ovidwhiech asquahownstum esthat thteimcaotrorersuposinndginp( 1r/2ise) problem is BQP-complete and improves the QMA upper bound sketched by Liu and Wang (ESA 2025) . Together with an Ω(1/ϵ) quantum query complexity lower bound, this leaves only square-root room for improvement.  \nThe key intuition behind our estimators is that, when one state is pure, the pure state |ψ⟩ has overlap at least 1/2 with the top unit eigenvector of |ψ⟩⟨ψ| − ρ, reflecting a structural feature specific to the operator norm distance.  \n∗ Email: [yupan.liu@epfl.ch](yupan.liu@epfl.ch)  \n†Email: [QishengWang1994@gmail.com](QishengWang1994@gmail.com)  \n‡Email: [yu.zhan@u.nus.edu](yu.zhan@u.nus.edu)  \nContents  \n1 Introduction 1  \n1. 1 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2  \n1.2 Proof techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3  \n1.3 Discussion and open problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5  \n1.4 Related works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6  \n2 Preliminaries 6  \n2. 1 Closeness measures for quantum states . . . . . . . . . . . . . . . . . . . . . . . . 6  \n2.2 Closeness testing of quantum states via state-preparation circuits . . . . . . . . . 7  \n2.3 Unitary dilation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8  \n2.4 Quantum sample-to-query lifting ........................... 8  \n3 Efficient quantum algorithms for estimating trace and operator norm distances when one state is pure 9  \n3.1 The input states provide a warm start when one state is pure . . . . . . . . . . . 9  \n3.2 Qubitization for unitary dilation ........................... 10  \n3.3 Quantum query algorithm when one state is pure .................. 12  \n3.3.1 The quantum query algorithm . . . . . . . . . . . . . . . . . . . . . . . . 12  \n3.4 Quantum sample algorithm when one state is pure ................. 15  \n4 Efficient quantum algorithms for estimating operator norm distance 16  \n4.1 The input states provide an eigenvalue-scaled overlap . . . . . . . . . . . . . . . . 16  \n4.2 Quantum query algorithm for estimating operator norm distance ......... 17  \n4.3 Quantum sample algorithm for estimating operator norm distance ........ 20","cbCaihcBbZNiRE5f","https://ap.wps.com/l/cbCaihcBbZNiRE5f","pdf",749235,6,1,28,"English","en",105,"# Contents\n## Introduction\n## Main results\n## Proof techniques\n## Discussion and open problems\n## Related works\n## Preliminaries\n## Closeness measures for quantum states\n## Efficient quantum algorithms when one state is pure\n## Efficient quantum algorithms for estimating operator norm distance","[{\"question\":\"What distance measures are studied for quantum states?\",\"answer\":\"The document focuses on the operator norm distance T∞(ρ0,ρ1)=∥ρ0−ρ1∥∞, where ∥A∥∞ is the largest singular value. It also relates the operator norm distance to trace distance when one state is pure.\"},{\"question\":\"What is the key complexity result when one state is pure?\",\"answer\":\"An optimal quantum estimator is provided using Θ(1/ε) queries to the state-preparation circuits. For constant additive error (e.g., ε=1/5), the runtime is poly(n), and the query complexity becomes independent of the rank/dimension of the mixed state.\"},{\"question\":\"How does the work compare to prior approaches for estimating T∞ or trace distance?\",\"answer\":\"The document states that prior methods (e.g., van Apeldoorn, Cornelissen, Gilyén, and Nannicini; and Wang and Zhang) have query complexity scaling at least linearly with rank(ρ), which can be exponential in general. The new approach matches optimal bounds when both states are pure, and improves rank dependence in the pure/mixed setting.\"},{\"question\":\"What intuition is used to design the estimators?\",\"answer\":\"When one state is pure, the pure state |ψ⟩ has overlap at least 1/2 with the top unit eigenvector of |ψ⟩⟨ψ|−ρ, reflecting a structural feature specific to the operator norm distance.\"}]",1784190242,71,{"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":91,"head_meta":93,"extra_data":95,"updated_unix":28},"on-estimating-operator-norm-distance-with-optimal-trace-distance-estimation-when-one-state-is-pure","",{"@graph":36,"@context":90},[37,54,69],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,51],{"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":50},"https://docshare.wps.com/document/research-report/",3,{"item":52,"name":13,"@type":43,"position":53},"https://docshare.wps.com/document/on-estimating-operator-norm-distance-with-optimal-trace-distance-estimation-when-one-state-is-pure/83757/",4,{"url":52,"name":13,"@type":55,"author":56,"headline":13,"publisher":58,"fileFormat":61,"inLanguage":24,"description":14,"dateModified":62,"datePublished":63,"encodingFormat":61,"isAccessibleForFree":64,"interactionStatistic":65},"DigitalDocument",{"name":9,"@type":57},"Person",{"url":41,"name":59,"@type":60},"DocShare","Organization","application/pdf","2026-07-27","2026-07-16",true,{"@type":66,"interactionType":67,"userInteractionCount":20},"InteractionCounter",{"@type":68},"ViewAction",{"@type":70,"mainEntity":71},"FAQPage",[72,78,82,86],{"name":73,"@type":74,"acceptedAnswer":75},"What distance measures are studied for quantum states?","Question",{"text":76,"@type":77},"The document focuses on the operator norm distance T∞(ρ0,ρ1)=∥ρ0−ρ1∥∞, where ∥A∥∞ is the largest singular value. It also relates the operator norm distance to trace distance when one state is pure.","Answer",{"name":79,"@type":74,"acceptedAnswer":80},"What is the key complexity result when one state is pure?",{"text":81,"@type":77},"An optimal quantum estimator is provided using Θ(1/ε) queries to the state-preparation circuits. For constant additive error (e.g., ε=1/5), the runtime is poly(n), and the query complexity becomes independent of the rank/dimension of the mixed state.",{"name":83,"@type":74,"acceptedAnswer":84},"How does the work compare to prior approaches for estimating T∞ or trace distance?",{"text":85,"@type":77},"The document states that prior methods (e.g., van Apeldoorn, Cornelissen, Gilyén, and Nannicini; and Wang and Zhang) have query complexity scaling at least linearly with rank(ρ), which can be exponential in general. The new approach matches optimal bounds when both states are pure, and improves rank dependence in the pure/mixed setting.",{"name":87,"@type":74,"acceptedAnswer":88},"What intuition is used to design the estimators?",{"text":89,"@type":77},"When one state is pure, the pure state |ψ⟩ has overlap at least 1/2 with the top unit eigenvector of |ψ⟩⟨ψ|−ρ, reflecting a structural feature specific to the operator norm distance.","https://schema.org",{"og:url":52,"og:type":92,"og:title":13,"og:site_name":59,"og:description":14},"article",{"robots":94,"canonical":52},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":97},[98,102,106,110,115,119,124,127,132,135,139],{"id":21,"doc_module":4,"doc_module_name":46,"category_name":99,"show_sort_weight":100,"slug":101},"Story & Novel",90,"story-novel",{"id":47,"doc_module":4,"doc_module_name":46,"category_name":103,"show_sort_weight":104,"slug":105},"Literature",80,"literature",{"id":53,"doc_module":4,"doc_module_name":46,"category_name":107,"show_sort_weight":108,"slug":109},"Exam",70,"exam",{"id":111,"doc_module":4,"doc_module_name":46,"category_name":112,"show_sort_weight":113,"slug":114},5,"Comic",60,"comic",{"id":20,"doc_module":4,"doc_module_name":46,"category_name":116,"show_sort_weight":117,"slug":118},"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":111,"slug":142},19,"General","general"]