[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-81846-en":3,"doc-seo-81846-105":31,"detail-sidebar-cat-0-en-105":93},{"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":28,"seo_description":14,"update_tm":29,"read_time":30},81846,5909877438554,"Maeve","https://ap-avatar.wpscdn.com/avatar/5600025385ad2bf12a7?_k=1778553567797529272",8,"Research & Report","Optimal Stabilizer Testing and Learning with Limited Quantum Memory","Studies stabilizer state testing and learning when an algorithm has only limited coherent quantum memory between measurements. A sequential procedure receives copies of an unknown n-qubit state while retaining at most k coherent qubits. With unlimited memory, testing uses a constant number of copies independent of dimension, unlike learning. Under memory constraints, the testing-vs-learning separation collapses: testing requires Θ(n−k) copies, and nonadaptive learning requires Θ(n²/k) copies. An application yields exponential lower bounds for purity testing and identifies coherent quantum memory as the key resource enabling the separation.","Optimal stabilizer testing and learning with limited quantum memory  \narXiv :2607 .02444v1 [ quant-ph] 2 Jul 2026  \nSrinivasan Arunachalam  \nIBM Research  \nSilicon Valley [Srinivasan.Arunachalam@ibm.com](Srinivasan.Arunachalam@ibm.com)  \nLouis Schatzki  \nDahlem Center for Complex Quantum Systems Freie Universit¨at Berlin [real-louismares98@zedat.fu-berlin.de](real-louismares98@zedat.fu-berlin.de)  \nJuly 3, 2026  \nAbstract  \nWe study stabilizer state testing and learning with limited coherent quantum memory. Here an algorithm sequentially receives copies of an unknown n-qubit state, but may keep only k qubits of coherent quantum memory between measurements. With unrestricted memory, seminal work of Gross, Nezami and Walter [GNW21] showed how to test n-qubit stabilizer states using 6 copies, which is dimension independent, unlike the learning complexity of Θ(n) . We show that this testing-vs-learning separation is lost under memory constraints. More concretely we show that  \n1. The sample complexity of testing stabilizer states in the k-qubit memory framework is Θ(n − k) . Our upper bound goes via a novel connection to the hidden shift problem and the lower bound is proven using a novel approach to average case bounds on likelihood ratios via combinatorics of the stochastic orthogonal group.  \n2. The sample complexity of learning stabilizer states with k qubits of memory, in the nonadaptive framework, is Θ(n2 /k) .  \nAs a further application of our techniques, we prove an exponential lower bound for purity testing even when the memory may be left coherent throughout the protocol. Our main results identify coherent quantum memory as the resource enabling the usual separation between stabilizer testing and learning. In particular, even with k = 0 .99n qubits of memory, there is no constantcopy stabilizer tester; furthermore for k = cn qubits of memory (for 0 \u003C c \u003C 1), stabilizer testing is as hard as learning, with both requiring Θ(n) copies.  \nContents  \n1 Introduction 3  \n1.1 Main results ................................................... 4  \n1.2 Proof sketch of testing upper bound ..................................... 6  \n1.3 Proof sketch of testing lower bound ...................................... 8  \n1.4 Proof sketch of learning bounds ........................................ 10  \n1.5 Discussion and open questions ......................................... 11  \n2 Preliminaries 13  \n2.1 Paulis and Cliffords ............................................... 13  \n2.2 Fourier analysis ................................................. 15  \n2.3 Stochastic orthogonal group ......................................... 16  \n3 Technical toolkit 17  \n3.1 Partial bell sampling .............................................. 17  \n3.2 Ensembles corresponding to deg-2 phase states ............................... 20  \n3.3 Structure of protocols with quantum memory ................................ 21  \n4 Testing upper bound 24  \n4.1 Partial Bell sampling .............................................. 24  \n4.2 The hidden-shift subroutine .......................................... 24  \n4.3 The final tester ................................................. 29  \n4.4 Conditional soundness of the graph test ................................... 30  \n4.5 Global soundness: finding a bad prefix .................................... 32  \n5 Testing lower bounds 36  \n5.1 Information Theoretic Lower Bounds ..................................... 36  \n5.2 Construction of hard ensemble ........................................ 38  \n5.3 Rewriting our phase state ensemble ...................................... 39  \n5.4 Lower bound for testing ............................................ 41  \n5.5 Proof of Lemma 5.8 .............................................. 44  \n6 Non-adaptive learning stabilizer states 52  \n6.1 Upper bounds .................................................. 52  \n6.2 Lower bound .................................................. 56  \nA Total variatio","cbCairfAGsJfgYOU","https://ap.wps.com/l/cbCairfAGsJfgYOU","pdf",2800009,5,1,66,"English","en",105,"# Introduction\n## Main results\n## Proof sketch of testing upper bound\n## Proof sketch of testing lower bound\n## Proof sketch of learning bounds\n## Discussion and open questions\n# Preliminaries\n## Paulis and Cliffords\n## Fourier analysis\n## Stochastic orthogonal group\n# Technical toolkit\n## Partial bell sampling\n## Ensembles corresponding to deg-2 phase states\n## Structure of protocols with quantum memory\n# Testing upper bound\n## Partial Bell sampling\n## The hidden-shift subroutine\n## The final tester\n## Conditional soundness of the graph test\n## Global soundness: finding a bad prefix\n# Testing lower bounds\n## Information Theoretic Lower Bounds\n## Construction of hard ensemble\n## Rewriting our phase state ensemble\n## Lower bound for testing\n## Proof of Lemma 5.8\n# Non-adaptive learning stabilizer states\n## Upper bounds\n## Lower bound\n# Purity Testing Lower Bound","[{\"question\":\"What problem does the document address?\",\"answer\":\"It analyzes how stabilizer-state testing and learning change when the algorithm is restricted to storing only k qubits coherently between measurements.\"},{\"question\":\"How does limited coherent quantum memory affect sample complexity?\",\"answer\":\"Testing stabilizer states requires Θ(n−k) copies, and nonadaptive learning requires Θ(n²/k) copies, showing a loss of the usual testing-vs-learning separation.\"},{\"question\":\"What resource enables the separation between stabilizer testing and learning?\",\"answer\":\"Coherent quantum memory: the results indicate that without enough coherent memory, testing becomes as hard as learning, and even near-linear memory does not yield a constant-copy stabilizer tester.\"}]","Optimal Stabilizer Testing and Learning with Limited Quantum Memory | PDF",1784176612,166,{"code":4,"msg":32,"data":33},"ok",{"site_id":25,"language":24,"slug":34,"title":13,"keywords":35,"description":14,"schema_data":36,"social_meta":88,"head_meta":90,"extra_data":92,"updated_unix":29},"optimal-stabilizer-testing-and-learning-with-limited-quantum-memory","",{"@graph":37,"@context":87},[38,55,70],{"@type":39,"itemListElement":40},"BreadcrumbList",[41,45,49,52],{"item":42,"name":43,"@type":44,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":46,"name":47,"@type":44,"position":48},"https://docshare.wps.com/document/","Document",2,{"item":50,"name":12,"@type":44,"position":51},"https://docshare.wps.com/document/research-report/",3,{"item":53,"name":13,"@type":44,"position":54},"https://docshare.wps.com/document/optimal-stabilizer-testing-and-learning-with-limited-quantum-memory/81846/",4,{"url":53,"name":13,"@type":56,"author":57,"headline":13,"publisher":59,"fileFormat":62,"inLanguage":24,"description":14,"dateModified":63,"datePublished":64,"encodingFormat":62,"isAccessibleForFree":65,"interactionStatistic":66},"DigitalDocument",{"name":9,"@type":58},"Person",{"url":42,"name":60,"@type":61},"DocShare","Organization","application/pdf","2026-07-29","2026-07-16",true,{"@type":67,"interactionType":68,"userInteractionCount":20},"InteractionCounter",{"@type":69},"ViewAction",{"@type":71,"mainEntity":72},"FAQPage",[73,79,83],{"name":74,"@type":75,"acceptedAnswer":76},"What problem does the document address?","Question",{"text":77,"@type":78},"It analyzes how stabilizer-state testing and learning change when the algorithm is restricted to storing only k qubits coherently between measurements.","Answer",{"name":80,"@type":75,"acceptedAnswer":81},"How does limited coherent quantum memory affect sample complexity?",{"text":82,"@type":78},"Testing stabilizer states requires Θ(n−k) copies, and nonadaptive learning requires Θ(n²/k) copies, showing a loss of the usual testing-vs-learning separation.",{"name":84,"@type":75,"acceptedAnswer":85},"What resource enables the separation between stabilizer testing and learning?",{"text":86,"@type":78},"Coherent quantum memory: the results indicate that without enough coherent memory, testing becomes as hard as learning, and even near-linear memory does not yield a constant-copy stabilizer tester.","https://schema.org",{"og:url":53,"og:type":89,"og:title":13,"og:site_name":60,"og:description":14},"article",{"robots":91,"canonical":53},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":94},[95,99,103,107,111,116,121,124,129,132,136],{"id":21,"doc_module":4,"doc_module_name":47,"category_name":96,"show_sort_weight":97,"slug":98},"Story & Novel",90,"story-novel",{"id":48,"doc_module":4,"doc_module_name":47,"category_name":100,"show_sort_weight":101,"slug":102},"Literature",80,"literature",{"id":54,"doc_module":4,"doc_module_name":47,"category_name":104,"show_sort_weight":105,"slug":106},"Exam",70,"exam",{"id":20,"doc_module":4,"doc_module_name":47,"category_name":108,"show_sort_weight":109,"slug":110},"Comic",60,"comic",{"id":112,"doc_module":4,"doc_module_name":47,"category_name":113,"show_sort_weight":114,"slug":115},6,"Technology",50,"technology",{"id":117,"doc_module":4,"doc_module_name":47,"category_name":118,"show_sort_weight":119,"slug":120},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":47,"category_name":12,"show_sort_weight":122,"slug":123},30,"research-report",{"id":125,"doc_module":4,"doc_module_name":47,"category_name":126,"show_sort_weight":127,"slug":128},9,"Religion & Spirituality",20,"religion-spirituality",{"id":127,"doc_module":4,"doc_module_name":47,"category_name":130,"show_sort_weight":127,"slug":131},"World Cup","world-cup",{"id":133,"doc_module":4,"doc_module_name":47,"category_name":134,"show_sort_weight":133,"slug":135},10,"Lifestyle","lifestyle",{"id":137,"doc_module":4,"doc_module_name":47,"category_name":138,"show_sort_weight":20,"slug":139},19,"General","general"]