{"id":"https://openalex.org/W4391406967","doi":"https://doi.org/10.1109/tit.2024.3360951","title":"Quantum State Tomography for Matrix Product Density Operators","display_name":"Quantum State Tomography for Matrix Product Density Operators","publication_year":2024,"publication_date":"2024-01-31","ids":{"openalex":"https://openalex.org/W4391406967","doi":"https://doi.org/10.1109/tit.2024.3360951"},"language":"en","primary_location":{"id":"doi:10.1109/tit.2024.3360951","is_oa":false,"landing_page_url":"https://doi.org/10.1109/tit.2024.3360951","pdf_url":null,"source":{"id":"https://openalex.org/S4502562","display_name":"IEEE Transactions on Information Theory","issn_l":"0018-9448","issn":["0018-9448","1557-9654"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319808","host_organization_name":"Institute of Electrical and Electronics Engineers","host_organization_lineage":["https://openalex.org/P4310319808"],"host_organization_lineage_names":["Institute of Electrical and Electronics Engineers"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"IEEE Transactions on Information Theory","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":false,"oa_status":"closed","oa_url":null,"any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5100742403","display_name":"Zhen Qin","orcid":"https://orcid.org/0009-0000-0388-8840"},"institutions":[{"id":"https://openalex.org/I52357470","display_name":"The Ohio State University","ror":"https://ror.org/00rs6vg23","country_code":"US","type":"education","lineage":["https://openalex.org/I52357470"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Zhen Qin","raw_affiliation_strings":["Department of Computer Science and Engineering, The Ohio State University, Columbus, OH, USA"],"raw_orcid":"https://orcid.org/0009-0000-0388-8840","affiliations":[{"raw_affiliation_string":"Department of Computer Science and Engineering, The Ohio State University, Columbus, OH, USA","institution_ids":["https://openalex.org/I52357470"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5062636466","display_name":"Casey Jameson","orcid":"https://orcid.org/0000-0002-1428-4556"},"institutions":[{"id":"https://openalex.org/I167576493","display_name":"Colorado School of Mines","ror":"https://ror.org/04raf6v53","country_code":"US","type":"education","lineage":["https://openalex.org/I167576493"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Casey Jameson","raw_affiliation_strings":["Department of Physics, Colorado School of Mines, Golden, CO, USA","Department of Physics, Colorado School of Mines, Golden, Colorado, USA"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Physics, Colorado School of Mines, Golden, CO, USA","institution_ids":["https://openalex.org/I167576493"]},{"raw_affiliation_string":"Department of Physics, Colorado School of Mines, Golden, Colorado, USA","institution_ids":["https://openalex.org/I167576493"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5087022291","display_name":"Zhexuan Gong","orcid":"https://orcid.org/0000-0001-9018-7865"},"institutions":[{"id":"https://openalex.org/I167576493","display_name":"Colorado School of Mines","ror":"https://ror.org/04raf6v53","country_code":"US","type":"education","lineage":["https://openalex.org/I167576493"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Zhexuan Gong","raw_affiliation_strings":["Department of Physics, Colorado School of Mines, Golden, CO, USA","Department of Physics, Colorado School of Mines, Golden, Colorado, USA"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Physics, Colorado School of Mines, Golden, CO, USA","institution_ids":["https://openalex.org/I167576493"]},{"raw_affiliation_string":"Department of Physics, Colorado School of Mines, Golden, Colorado, USA","institution_ids":["https://openalex.org/I167576493"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5065207980","display_name":"Michael B. Wakin","orcid":"https://orcid.org/0000-0002-2165-4586"},"institutions":[{"id":"https://openalex.org/I167576493","display_name":"Colorado School of Mines","ror":"https://ror.org/04raf6v53","country_code":"US","type":"education","lineage":["https://openalex.org/I167576493"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Michael B. Wakin","raw_affiliation_strings":["Department of Electrical Engineering, Colorado School of Mines, Golden, CO, USA","Department of Electrical Engineering, Colorado School of Mines, Golden, Colorado, USA"],"raw_orcid":"https://orcid.org/0000-0002-2165-4586","affiliations":[{"raw_affiliation_string":"Department of Electrical Engineering, Colorado School of Mines, Golden, CO, USA","institution_ids":["https://openalex.org/I167576493"]},{"raw_affiliation_string":"Department of Electrical Engineering, Colorado School of Mines, Golden, Colorado, USA","institution_ids":["https://openalex.org/I167576493"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5011989964","display_name":"Zhihui Zhu","orcid":"https://orcid.org/0000-0002-3856-0375"},"institutions":[{"id":"https://openalex.org/I52357470","display_name":"The Ohio State University","ror":"https://ror.org/00rs6vg23","country_code":"US","type":"education","lineage":["https://openalex.org/I52357470"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Zhihui Zhu","raw_affiliation_strings":["Department of Computer Science and Engineering, The Ohio State University, Columbus, OH, USA"],"raw_orcid":"https://orcid.org/0000-0002-3856-0375","affiliations":[{"raw_affiliation_string":"Department of Computer Science and Engineering, The Ohio State University, Columbus, OH, USA","institution_ids":["https://openalex.org/I52357470"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":4.1677,"has_fulltext":false,"cited_by_count":15,"citation_normalized_percentile":{"value":0.94478726,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":90,"max":99},"biblio":{"volume":"70","issue":"7","first_page":"5030","last_page":"5056"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10020","display_name":"Quantum Information and Cryptography","score":0.9997000098228455,"subfield":{"id":"https://openalex.org/subfields/1702","display_name":"Artificial Intelligence"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T10020","display_name":"Quantum Information and Cryptography","score":0.9997000098228455,"subfield":{"id":"https://openalex.org/subfields/1702","display_name":"Artificial Intelligence"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10682","display_name":"Quantum Computing Algorithms and Architecture","score":0.9995999932289124,"subfield":{"id":"https://openalex.org/subfields/1702","display_name":"Artificial Intelligence"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10500","display_name":"Sparse and Compressive Sensing Techniques","score":0.9994999766349792,"subfield":{"id":"https://openalex.org/subfields/2206","display_name":"Computational Mechanics"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/quantum-tomography","display_name":"Quantum tomography","score":0.7670057415962219},{"id":"https://openalex.org/keywords/quantum-state","display_name":"Quantum state","score":0.616247296333313},{"id":"https://openalex.org/keywords/dimension","display_name":"Dimension (graph theory)","score":0.4978048801422119},{"id":"https://openalex.org/keywords/qubit","display_name":"Qubit","score":0.4831406772136688},{"id":"https://openalex.org/keywords/quantum-information","display_name":"Quantum information","score":0.4749436676502228},{"id":"https://openalex.org/keywords/quantum","display_name":"Quantum","score":0.4601256549358368},{"id":"https://openalex.org/keywords/quantum-entanglement","display_name":"Quantum entanglement","score":0.459859162569046},{"id":"https://openalex.org/keywords/quantum-computer","display_name":"Quantum computer","score":0.4552924931049347},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.44701117277145386},{"id":"https://openalex.org/keywords/quantum-algorithm","display_name":"Quantum algorithm","score":0.4443582594394684},{"id":"https://openalex.org/keywords/density-matrix","display_name":"Density matrix","score":0.44088014960289},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.39837148785591125},{"id":"https://openalex.org/keywords/statistical-physics","display_name":"Statistical physics","score":0.38306689262390137},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.35943442583084106},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.34546661376953125},{"id":"https://openalex.org/keywords/quantum-mechanics","display_name":"Quantum mechanics","score":0.29001733660697937},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.20870810747146606},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.19455993175506592}],"concepts":[{"id":"https://openalex.org/C166834278","wikidata":"https://www.wikidata.org/wiki/Q4218494","display_name":"Quantum tomography","level":4,"score":0.7670057415962219},{"id":"https://openalex.org/C15706264","wikidata":"https://www.wikidata.org/wiki/Q230883","display_name":"Quantum state","level":3,"score":0.616247296333313},{"id":"https://openalex.org/C33676613","wikidata":"https://www.wikidata.org/wiki/Q13415176","display_name":"Dimension (graph theory)","level":2,"score":0.4978048801422119},{"id":"https://openalex.org/C203087015","wikidata":"https://www.wikidata.org/wiki/Q378201","display_name":"Qubit","level":3,"score":0.4831406772136688},{"id":"https://openalex.org/C169699857","wikidata":"https://www.wikidata.org/wiki/Q2122243","display_name":"Quantum information","level":3,"score":0.4749436676502228},{"id":"https://openalex.org/C84114770","wikidata":"https://www.wikidata.org/wiki/Q46344","display_name":"Quantum","level":2,"score":0.4601256549358368},{"id":"https://openalex.org/C121040770","wikidata":"https://www.wikidata.org/wiki/Q215675","display_name":"Quantum entanglement","level":3,"score":0.459859162569046},{"id":"https://openalex.org/C58053490","wikidata":"https://www.wikidata.org/wiki/Q176555","display_name":"Quantum computer","level":3,"score":0.4552924931049347},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.44701117277145386},{"id":"https://openalex.org/C137019171","wikidata":"https://www.wikidata.org/wiki/Q2623817","display_name":"Quantum algorithm","level":3,"score":0.4443582594394684},{"id":"https://openalex.org/C56911000","wikidata":"https://www.wikidata.org/wiki/Q831774","display_name":"Density matrix","level":3,"score":0.44088014960289},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.39837148785591125},{"id":"https://openalex.org/C121864883","wikidata":"https://www.wikidata.org/wiki/Q677916","display_name":"Statistical physics","level":1,"score":0.38306689262390137},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.35943442583084106},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.34546661376953125},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.29001733660697937},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.20870810747146606},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.19455993175506592}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1109/tit.2024.3360951","is_oa":false,"landing_page_url":"https://doi.org/10.1109/tit.2024.3360951","pdf_url":null,"source":{"id":"https://openalex.org/S4502562","display_name":"IEEE Transactions on Information Theory","issn_l":"0018-9448","issn":["0018-9448","1557-9654"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319808","host_organization_name":"Institute of Electrical and Electronics Engineers","host_organization_lineage":["https://openalex.org/P4310319808"],"host_organization_lineage_names":["Institute of Electrical and Electronics Engineers"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"IEEE Transactions on Information Theory","raw_type":"journal-article"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G1028307773","display_name":null,"funder_award_id":"CCF-2106834","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G2202711852","display_name":null,"funder_award_id":"PHY-2112893","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G2656338998","display_name":null,"funder_award_id":"CCF-2241298","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G3464499140","display_name":null,"funder_award_id":"CCF-1839232","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"}],"funders":[{"id":"https://openalex.org/F4320306076","display_name":"National Science Foundation","ror":"https://ror.org/021nxhr62"},{"id":"https://openalex.org/F4320306159","display_name":"W. M. Keck Foundation","ror":"https://ror.org/000dswa46"}],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":154,"referenced_works":["https://openalex.org/W966126569","https://openalex.org/W1480812382","https://openalex.org/W1509689762","https://openalex.org/W1550516781","https://openalex.org/W1551092272","https://openalex.org/W1559981310","https://openalex.org/W1563049925","https://openalex.org/W1631356911","https://openalex.org/W1684009677","https://openalex.org/W1853666739","https://openalex.org/W1922629554","https://openalex.org/W1942002803","https://openalex.org/W1965471276","https://openalex.org/W1975861455","https://openalex.org/W1993482030","https://openalex.org/W2001518794","https://openalex.org/W2014237348","https://openalex.org/W2016258127","https://openalex.org/W2025533349","https://openalex.org/W2025842401","https://openalex.org/W2030449718","https://openalex.org/W2036133196","https://openalex.org/W2036510582","https://openalex.org/W2037164428","https://openalex.org/W2039312417","https://openalex.org/W2066972040","https://openalex.org/W2089817321","https://openalex.org/W2096391265","https://openalex.org/W2098271372","https://openalex.org/W2100556411","https://openalex.org/W2103775046","https://openalex.org/W2104608642","https://openalex.org/W2117790027","https://openalex.org/W2118550318","https://openalex.org/W2119667497","https://openalex.org/W2119745496","https://openalex.org/W2122816100","https://openalex.org/W2122932035","https://openalex.org/W2123338985","https://openalex.org/W2124492577","https://openalex.org/W2131172946","https://openalex.org/W2131841551","https://openalex.org/W2134348739","https://openalex.org/W2134793668","https://openalex.org/W2145096794","https://openalex.org/W2148701549","https://openalex.org/W2162451874","https://openalex.org/W2166949820","https://openalex.org/W2167077875","https://openalex.org/W2180180824","https://openalex.org/W2202775496","https://openalex.org/W2286769582","https://openalex.org/W2290336547","https://openalex.org/W2412278696","https://openalex.org/W2431890537","https://openalex.org/W2521692805","https://openalex.org/W2564982350","https://openalex.org/W2565722603","https://openalex.org/W2619959423","https://openalex.org/W2649051464","https://openalex.org/W2726993892","https://openalex.org/W2765932895","https://openalex.org/W2781738013","https://openalex.org/W2783318359","https://openalex.org/W2795485111","https://openalex.org/W2884037738","https://openalex.org/W2887054207","https://openalex.org/W2890107213","https://openalex.org/W2920652573","https://openalex.org/W2922946882","https://openalex.org/W2923537029","https://openalex.org/W2945150561","https://openalex.org/W2949757588","https://openalex.org/W2962708900","https://openalex.org/W2963116674","https://openalex.org/W2963293339","https://openalex.org/W2963441888","https://openalex.org/W2963544062","https://openalex.org/W2963583445","https://openalex.org/W2971963065","https://openalex.org/W2982169647","https://openalex.org/W3005852419","https://openalex.org/W3006823398","https://openalex.org/W3007765404","https://openalex.org/W3010571098","https://openalex.org/W3011222909","https://openalex.org/W3013002783","https://openalex.org/W3013490000","https://openalex.org/W3080203339","https://openalex.org/W3084875062","https://openalex.org/W3086385056","https://openalex.org/W3097999675","https://openalex.org/W3098737781","https://openalex.org/W3100802578","https://openalex.org/W3100931082","https://openalex.org/W3101491133","https://openalex.org/W3102636246","https://openalex.org/W3103713775","https://openalex.org/W3103870741","https://openalex.org/W3104065018","https://openalex.org/W3104184357","https://openalex.org/W3104481216","https://openalex.org/W3104744854","https://openalex.org/W3104774502","https://openalex.org/W3105213343","https://openalex.org/W3123208069","https://openalex.org/W3175418406","https://openalex.org/W3176786781","https://openalex.org/W3196385563","https://openalex.org/W3205245241","https://openalex.org/W3212210483","https://openalex.org/W4226096164","https://openalex.org/W4231187174","https://openalex.org/W4245577611","https://openalex.org/W4250954493","https://openalex.org/W4250955649","https://openalex.org/W4285483469","https://openalex.org/W4286908170","https://openalex.org/W4295547509","https://openalex.org/W4296932851","https://openalex.org/W4297672650","https://openalex.org/W4298155380","https://openalex.org/W4301425558","https://openalex.org/W4310576936","https://openalex.org/W4311094711","https://openalex.org/W4388041940","https://openalex.org/W4390689979","https://openalex.org/W6608343466","https://openalex.org/W6632935555","https://openalex.org/W6633475917","https://openalex.org/W6633544067","https://openalex.org/W6636505171","https://openalex.org/W6637178627","https://openalex.org/W6638060716","https://openalex.org/W6679168169","https://openalex.org/W6717575008","https://openalex.org/W6726590743","https://openalex.org/W6729942461","https://openalex.org/W6741042017","https://openalex.org/W6742193618","https://openalex.org/W6745442365","https://openalex.org/W6745544366","https://openalex.org/W6747544394","https://openalex.org/W6764409202","https://openalex.org/W6798699559","https://openalex.org/W6800258555","https://openalex.org/W6803246901","https://openalex.org/W6810611376","https://openalex.org/W6839457639","https://openalex.org/W6839476194","https://openalex.org/W6839572662","https://openalex.org/W6842877371","https://openalex.org/W6847487651","https://openalex.org/W6860784962"],"related_works":["https://openalex.org/W3100430039","https://openalex.org/W1964028587","https://openalex.org/W2298044093","https://openalex.org/W3157044737","https://openalex.org/W2623245987","https://openalex.org/W2038389955","https://openalex.org/W2015513129","https://openalex.org/W4387560890","https://openalex.org/W4366835629","https://openalex.org/W4386468892"],"abstract_inverted_index":{"The":[0],"reconstruction":[1],"of":[2,22,38,49,100,103,137,151,170,180,189,220,232,238,269,275,284,313,344],"quantum":[3,11,23,32,65,74,258,336,351],"states":[4,69,83,123,144],"from":[5,174],"experimental":[6],"measurements,":[7],"often":[8],"achieved":[9,293],"using":[10,172,217],"state":[12,33,39,108,138,270,297],"tomography":[13],"(QST),":[14],"is":[15,112,277],"crucial":[16],"for":[17,28,56,121,141,166,348],"the":[18,47,53,57,101,135,149,167,178,203,230,239,273,296,305,311,342],"verification":[19],"and":[20,162,177,195],"benchmarking":[21],"devices.":[24],"However,":[25],"performing":[26],"QST":[27,117,245,346],"a":[29,96,210,218,264],"generic":[30],"unstructured":[31],"requires":[34],"an":[35,207,285],"enormous":[36],"number":[37,48,102,136,150,219,231,268,274],"copies":[40,139,271],"that":[41,145,202,223,251,262,328],"grows":[42],"<italic":[43,226,265],"xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"":[44,227,266],"xmlns:xlink=\"http://www.w3.org/1999/xlink\">exponentially</i>":[45],"with":[46,95,148,209,246,310],"individual":[50],"quanta":[51],"in":[52,124,206,253,272,298],"system,":[54],"even":[55],"most":[58],"optimal":[59],"measurement":[60,191],"settings.":[61],"Fortunately,":[62],"many":[63],"physical":[64],"states,":[66],"such":[67,82,289],"as":[68],"generated":[70],"by":[71,90,185,294],"noisy,":[72],"intermediate-scale":[73],"computers,":[75],"are":[76,84,329],"usually":[77],"structured.":[78],"In":[79,126,153],"one":[80],"dimension,":[81],"expected":[85],"to":[86,158,279,321,332],"be":[87,119,215,255,292,319],"well":[88],"approximated":[89],"matrix":[91],"product":[92],"operators":[93],"(MPOs)":[94],"matrix/bond":[97],"dimension":[98,213],"independent":[99],"qubits,":[104,233],"therefore":[105],"enabling":[106],"efficient":[107,116,345],"representation.":[109],"Nevertheless,":[110],"it":[111],"still":[113],"unclear":[114],"whether":[115],"can":[118,214,252,291],"performed":[120],"these":[122],"general.":[125],"other":[127,349],"words,":[128],"there":[129],"exist":[130],"no":[131,235],"rigorous":[132],"bounds":[133],"on":[134,229,257,334],"required":[140,278],"reconstructing":[142],"MPO":[143,208,286],"scales":[146],"polynomially":[147],"qubits.":[152],"this":[154,160],"paper,":[155],"we":[156],"attempt":[157],"bridge":[159],"gap":[161],"establish":[163],"theoretical":[164],"guarantees":[165],"stable":[168],"recovery":[169,282,290],"MPOs":[171],"tools":[173],"compressive":[175],"sensing":[176],"theory":[179],"empirical":[181],"processes.":[182],"We":[183,200,241,260],"begin":[184],"studying":[186],"two":[187],"types":[188],"random":[190,197,221,248,300,323],"settings:":[192],"Gaussian":[193],"measurements":[194,222,250,327],"Haar":[196,247],"projective":[198,249],"measurements.":[199,240],"show":[201],"information":[204],"contained":[205],"constant":[211],"bond":[212],"preserved":[216],"depends":[224],"only":[225,263,302],"xmlns:xlink=\"http://www.w3.org/1999/xlink\">linearly</i>":[228],"assuming":[234],"statistical":[236,307],"error":[237,283,308],"then":[242],"study":[243],"MPO-based":[244],"principle":[254],"implemented":[256],"computers.":[259,337],"prove":[261],"xmlns:xlink=\"http://www.w3.org/1999/xlink\">polynomial</i>":[267],"qubits":[276],"guarantee":[280],"bounded":[281],"state.":[287],"Remarkably,":[288],"measuring":[295],"each":[299,314],"basis":[301],"once,":[303],"despite":[304],"large":[306],"associated":[309],"outcome":[312],"measurement.":[315],"Our":[316],"work":[317],"may":[318,339],"generalized":[320],"accommodate":[322],"local":[324],"or":[325],"t-design":[326],"more":[330],"practical":[331],"implement":[333],"current":[335],"It":[338],"also":[340],"facilitate":[341],"discovery":[343],"methods":[347],"structured":[350],"states.":[352]},"counts_by_year":[{"year":2026,"cited_by_count":5},{"year":2025,"cited_by_count":9},{"year":2024,"cited_by_count":1}],"updated_date":"2025-11-06T03:46:38.306776","created_date":"2025-10-10T00:00:00"}
