{"id":"https://openalex.org/W1485941945","doi":"https://doi.org/10.22331/q-2022-03-17-668","title":"The Pursuit of Uniqueness: Extending Valiant-Vazirani Theorem to the Probabilistic and Quantum Settings","display_name":"The Pursuit of Uniqueness: Extending Valiant-Vazirani Theorem to the Probabilistic and Quantum Settings","publication_year":2022,"publication_date":"2022-03-17","ids":{"openalex":"https://openalex.org/W1485941945","doi":"https://doi.org/10.22331/q-2022-03-17-668","mag":"1485941945"},"language":"en","primary_location":{"id":"doi:10.22331/q-2022-03-17-668","is_oa":true,"landing_page_url":"https://doi.org/10.22331/q-2022-03-17-668","pdf_url":"https://quantum-journal.org/papers/q-2022-03-17-668/pdf/","source":{"id":"https://openalex.org/S4210226432","display_name":"Quantum","issn_l":"2521-327X","issn":["2521-327X"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310317900","host_organization_name":"Verein zur F\u00f6rderung des Open Access Publizierens in den Quantenwissenschaften","host_organization_lineage":["https://openalex.org/P4310317900"],"host_organization_lineage_names":["Verein zur F\u00f6rderung des Open Access Publizierens in den Quantenwissenschaften"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Quantum","raw_type":"journal-article"},"type":"article","indexed_in":["arxiv","crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://quantum-journal.org/papers/q-2022-03-17-668/pdf/","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5005794608","display_name":"Dorit Aharonov","orcid":"https://orcid.org/0000-0002-9378-8495"},"institutions":[{"id":"https://openalex.org/I197251160","display_name":"Hebrew University of Jerusalem","ror":"https://ror.org/03qxff017","country_code":"IL","type":"education","lineage":["https://openalex.org/I197251160"]}],"countries":["IL"],"is_corresponding":false,"raw_author_name":"Dorit Aharonov","raw_affiliation_strings":["School of Computer Science and Engineering, The Hebrew University, Jerusalem, Israel"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Computer Science and Engineering, The Hebrew University, Jerusalem, Israel","institution_ids":["https://openalex.org/I197251160"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5007913595","display_name":"Michael Ben-Or","orcid":"https://orcid.org/0000-0002-7787-9311"},"institutions":[{"id":"https://openalex.org/I197251160","display_name":"Hebrew University of Jerusalem","ror":"https://ror.org/03qxff017","country_code":"IL","type":"education","lineage":["https://openalex.org/I197251160"]}],"countries":["IL"],"is_corresponding":false,"raw_author_name":"Michael Ben-Or","raw_affiliation_strings":["School of Computer Science and Engineering, The Hebrew University, Jerusalem, Israel"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Computer Science and Engineering, The Hebrew University, Jerusalem, Israel","institution_ids":["https://openalex.org/I197251160"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5030940558","display_name":"Fernando G. S. L. Brand\u00e3o","orcid":"https://orcid.org/0000-0003-3866-9378"},"institutions":[{"id":"https://openalex.org/I122411786","display_name":"California Institute of Technology","ror":"https://ror.org/05dxps055","country_code":"US","type":"education","lineage":["https://openalex.org/I122411786"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Fernando G.S.L. Brand\u00e3o","raw_affiliation_strings":["AWS Center for Quantum Computing, Pasadena, CA 91125, USA","IQIM, California Institute of Technology, Pasadena, CA 91125, USA"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"AWS Center for Quantum Computing, Pasadena, CA 91125, USA","institution_ids":[]},{"raw_affiliation_string":"IQIM, California Institute of Technology, Pasadena, CA 91125, USA","institution_ids":["https://openalex.org/I122411786"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5089611363","display_name":"Or Sattath","orcid":"https://orcid.org/0000-0001-7567-3822"},"institutions":[{"id":"https://openalex.org/I124227911","display_name":"Ben-Gurion University of the Negev","ror":"https://ror.org/05tkyf982","country_code":"IL","type":"education","lineage":["https://openalex.org/I124227911"]}],"countries":["IL"],"is_corresponding":false,"raw_author_name":"Or Sattath","raw_affiliation_strings":["Computer Science Department, Ben-Gurion University of the Negev, Beer-Sheva, Israel"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Computer Science Department, Ben-Gurion University of the Negev, Beer-Sheva, Israel","institution_ids":["https://openalex.org/I124227911"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":4,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":{"value":200,"currency":"EUR","value_usd":215},"apc_paid":{"value":200,"currency":"EUR","value_usd":215},"fwci":1.2486,"has_fulltext":true,"cited_by_count":23,"citation_normalized_percentile":{"value":0.82245697,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":98},"biblio":{"volume":"6","issue":null,"first_page":"668","last_page":"668"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10682","display_name":"Quantum Computing Algorithms and Architecture","score":0.9998999834060669,"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/T10682","display_name":"Quantum Computing Algorithms and Architecture","score":0.9998999834060669,"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/T10020","display_name":"Quantum Information and Cryptography","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/T11804","display_name":"Quantum many-body systems","score":0.9995999932289124,"subfield":{"id":"https://openalex.org/subfields/3107","display_name":"Atomic and Molecular Physics, and Optics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/hamiltonian","display_name":"Hamiltonian (control theory)","score":0.6310606598854065},{"id":"https://openalex.org/keywords/ground-state","display_name":"Ground state","score":0.5794631242752075},{"id":"https://openalex.org/keywords/quantum","display_name":"Quantum","score":0.4834235906600952},{"id":"https://openalex.org/keywords/quantum-algorithm","display_name":"Quantum algorithm","score":0.48084530234336853},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.4797707498073578},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.46264398097991943},{"id":"https://openalex.org/keywords/uniqueness","display_name":"Uniqueness","score":0.439566433429718},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.41112589836120605},{"id":"https://openalex.org/keywords/quantum-mechanics","display_name":"Quantum mechanics","score":0.34279948472976685},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.33311882615089417},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.08983597159385681}],"concepts":[{"id":"https://openalex.org/C130787639","wikidata":"https://www.wikidata.org/wiki/Q5645293","display_name":"Hamiltonian (control theory)","level":2,"score":0.6310606598854065},{"id":"https://openalex.org/C69523127","wikidata":"https://www.wikidata.org/wiki/Q4480008","display_name":"Ground state","level":2,"score":0.5794631242752075},{"id":"https://openalex.org/C84114770","wikidata":"https://www.wikidata.org/wiki/Q46344","display_name":"Quantum","level":2,"score":0.4834235906600952},{"id":"https://openalex.org/C137019171","wikidata":"https://www.wikidata.org/wiki/Q2623817","display_name":"Quantum algorithm","level":3,"score":0.48084530234336853},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.4797707498073578},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.46264398097991943},{"id":"https://openalex.org/C2777021972","wikidata":"https://www.wikidata.org/wiki/Q22976830","display_name":"Uniqueness","level":2,"score":0.439566433429718},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.41112589836120605},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.34279948472976685},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.33311882615089417},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.08983597159385681},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.0}],"mesh":[],"locations_count":4,"locations":[{"id":"doi:10.22331/q-2022-03-17-668","is_oa":true,"landing_page_url":"https://doi.org/10.22331/q-2022-03-17-668","pdf_url":"https://quantum-journal.org/papers/q-2022-03-17-668/pdf/","source":{"id":"https://openalex.org/S4210226432","display_name":"Quantum","issn_l":"2521-327X","issn":["2521-327X"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310317900","host_organization_name":"Verein zur F\u00f6rderung des Open Access Publizierens in den Quantenwissenschaften","host_organization_lineage":["https://openalex.org/P4310317900"],"host_organization_lineage_names":["Verein zur F\u00f6rderung des Open Access Publizierens in den Quantenwissenschaften"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Quantum","raw_type":"journal-article"},{"id":"pmh:oai:arXiv.org:0810.4840","is_oa":true,"landing_page_url":"http://arxiv.org/abs/0810.4840","pdf_url":"https://arxiv.org/pdf/0810.4840","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},{"id":"pmh:oai:authors.library.caltech.edu:67457","is_oa":true,"landing_page_url":"https://authors.library.caltech.edu/67457/","pdf_url":"https://resolver.caltech.edu/CaltechAUTHORS:20160527-123057985","source":{"id":"https://openalex.org/S4306402161","display_name":"CaltechAUTHORS (California Institute of Technology)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I122411786","host_organization_name":"California Institute of Technology","host_organization_lineage":["https://openalex.org/I122411786"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"Article"},{"id":"pmh:oai:doaj.org/article:de101eaf6d0b429fab2099bf12e81b99","is_oa":true,"landing_page_url":"https://doaj.org/article/de101eaf6d0b429fab2099bf12e81b99","pdf_url":null,"source":{"id":"https://openalex.org/S4306401280","display_name":"DOAJ (DOAJ: Directory of Open Access Journals)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by-sa","license_id":"https://openalex.org/licenses/cc-by-sa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Quantum, Vol 6, p 668 (2022)","raw_type":"article"}],"best_oa_location":{"id":"doi:10.22331/q-2022-03-17-668","is_oa":true,"landing_page_url":"https://doi.org/10.22331/q-2022-03-17-668","pdf_url":"https://quantum-journal.org/papers/q-2022-03-17-668/pdf/","source":{"id":"https://openalex.org/S4210226432","display_name":"Quantum","issn_l":"2521-327X","issn":["2521-327X"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310317900","host_organization_name":"Verein zur F\u00f6rderung des Open Access Publizierens in den Quantenwissenschaften","host_organization_lineage":["https://openalex.org/P4310317900"],"host_organization_lineage_names":["Verein zur F\u00f6rderung des Open Access Publizierens in den Quantenwissenschaften"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Quantum","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G7295279419","display_name":null,"funder_award_id":"GR/S82176/0","funder_id":"https://openalex.org/F4320334627","funder_display_name":"Engineering and Physical Sciences Research Council"}],"funders":[{"id":"https://openalex.org/F4320320283","display_name":"Imperial College London","ror":"https://ror.org/041kmwe10"},{"id":"https://openalex.org/F4320321950","display_name":"Hebrew University of Jerusalem","ror":"https://ror.org/03qxff017"},{"id":"https://openalex.org/F4320334627","display_name":"Engineering and Physical Sciences Research Council","ror":"https://ror.org/0439y7842"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W1485941945.pdf","grobid_xml":"https://content.openalex.org/works/W1485941945.grobid-xml"},"referenced_works_count":60,"referenced_works":["https://openalex.org/W10965301","https://openalex.org/W17898807","https://openalex.org/W200635132","https://openalex.org/W1485941945","https://openalex.org/W1527197079","https://openalex.org/W1534413569","https://openalex.org/W1549606184","https://openalex.org/W1556128349","https://openalex.org/W1556171410","https://openalex.org/W1585991502","https://openalex.org/W1633109858","https://openalex.org/W1667728213","https://openalex.org/W1938232311","https://openalex.org/W2016984055","https://openalex.org/W2019692968","https://openalex.org/W2021736779","https://openalex.org/W2033304364","https://openalex.org/W2036265926","https://openalex.org/W2044504659","https://openalex.org/W2045590835","https://openalex.org/W2052207834","https://openalex.org/W2058172107","https://openalex.org/W2059442002","https://openalex.org/W2072381205","https://openalex.org/W2084019889","https://openalex.org/W2088303041","https://openalex.org/W2096390054","https://openalex.org/W2096825534","https://openalex.org/W2099102182","https://openalex.org/W2099111195","https://openalex.org/W2100827027","https://openalex.org/W2101577710","https://openalex.org/W2105313913","https://openalex.org/W2116015765","https://openalex.org/W2116123253","https://openalex.org/W2170608991","https://openalex.org/W2192149141","https://openalex.org/W2223534485","https://openalex.org/W2289838009","https://openalex.org/W2478708596","https://openalex.org/W2559726630","https://openalex.org/W2783493337","https://openalex.org/W2798707604","https://openalex.org/W2804172638","https://openalex.org/W2895634032","https://openalex.org/W2895657145","https://openalex.org/W2897996634","https://openalex.org/W2949059721","https://openalex.org/W2949757588","https://openalex.org/W2953849388","https://openalex.org/W2990961515","https://openalex.org/W3037108023","https://openalex.org/W3044211125","https://openalex.org/W3103091093","https://openalex.org/W3103223207","https://openalex.org/W3105218880","https://openalex.org/W3105960111","https://openalex.org/W4251659063","https://openalex.org/W4255333822","https://openalex.org/W4298227433"],"related_works":["https://openalex.org/W571281153","https://openalex.org/W2949403936","https://openalex.org/W4301247218","https://openalex.org/W3023806604","https://openalex.org/W2952139525","https://openalex.org/W2129818458","https://openalex.org/W4289551052","https://openalex.org/W2117721338","https://openalex.org/W1971479511","https://openalex.org/W2051909960"],"abstract_inverted_index":{"Valiant-Vazirani":[0],"showed":[1],"in":[2,32,111,124],"1985":[3],"\\cite{VV85}":[4],"that":[5,11,87,130,155,185],"solving":[6],"NP":[7,25,125,137],"with":[8,70],"the":[9,23,37,44,57,61,88,95,115,147,167,170,177],"promise":[10],"\"yes\"":[12],"instances":[13],"have":[14,54],"only":[15],"one":[16],"witness":[17],"is":[18,104,110,123,142],"powerful":[19],"enough":[20],"to":[21,36,43,114,136,169,176,189],"solve":[22],"entire":[24],"class":[26,178],"(under":[27],"randomized":[28,107,139],"reductions).We":[29],"are":[30],"interested":[31],"extending":[33],"this":[34],"result":[35,175],"quantum":[38,67],"setting.":[39],"We":[40,85],"prove":[41],"extensions":[42],"classes":[45],"Merlin-Arthur":[46],"MA":[47],"and":[48,75],"Quantum-Classical-Merlin-Arthur":[49],"QCMA":[50,132],"\\cite{AN02}.":[51],"Our":[52],"results":[53],"implications":[55],"for":[56],"complexity":[58],"of":[59,65,94,99,117,146,150,159,166,172],"approximating":[60],"ground":[62,73,96,148],"state":[63,74,97,149],"energy":[64,98],"a":[66,71,164,191],"local":[68,102,153],"Hamiltonian":[69,154],"unique":[72],"an":[76,173],"<mml:math":[77],"xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mrow":[78],"class=\"MJX-TeXAtom-ORD\"><mml:mtext":[79],"class=\"MJX-tex-mathit\"":[80],"mathvariant=\"italic\">inverse":[81],"polynomial</mml:mtext></mml:mrow></mml:math>":[82],"spectral":[83],"gap.":[84],"show":[86,184],"estimation":[89],"(to":[90],"within":[91],"polynomial":[92,192],"accuracy)":[93],"poly-gapped":[100,152],"1-D":[101,120],"Hamiltonians":[103],"QCMA-hard,":[105],"under":[106],"reductions.":[108],"This":[109],"stark":[112],"contrast":[113],"case":[116],"constant":[118],"gapped":[119],"Hamiltonians,":[121],"which":[122],"\\cite{Has07}.":[126],"Moreover,":[127],"it":[128],"shows":[129],"unless":[131],"can":[133],"be":[134],"reduced":[135],"by":[138],"reductions,":[140],"there":[141],"no":[143],"classical":[144],"description":[145],"every":[151],"allows":[156],"efficient":[157],"calculation":[158],"expectation":[160],"values.Finally,":[161],"we":[162,183],"discuss":[163],"few":[165],"obstacles":[168],"establishment":[171],"analogous":[174],"Quantum-Merlin-Arthur":[179],"(QMA).":[180],"In":[181],"particular,":[182],"random":[186],"projections":[187],"fail":[188],"provide":[190],"gap":[193],"between":[194],"two":[195],"witnesses.":[196]},"counts_by_year":[{"year":2025,"cited_by_count":2},{"year":2024,"cited_by_count":2},{"year":2022,"cited_by_count":5},{"year":2021,"cited_by_count":4},{"year":2019,"cited_by_count":1},{"year":2016,"cited_by_count":3},{"year":2014,"cited_by_count":1},{"year":2013,"cited_by_count":4},{"year":2012,"cited_by_count":1}],"updated_date":"2026-06-11T09:08:48.828518","created_date":"2016-06-24T00:00:00"}