{"id":"https://openalex.org/W4384108178","doi":"https://doi.org/10.22331/q-2023-07-12-1057","title":"Quantum Goemans-Williamson Algorithm with the Hadamard Test and Approximate Amplitude Constraints","display_name":"Quantum Goemans-Williamson Algorithm with the Hadamard Test and Approximate Amplitude Constraints","publication_year":2023,"publication_date":"2023-07-12","ids":{"openalex":"https://openalex.org/W4384108178","doi":"https://doi.org/10.22331/q-2023-07-12-1057"},"language":"en","primary_location":{"id":"doi:10.22331/q-2023-07-12-1057","is_oa":true,"landing_page_url":"https://doi.org/10.22331/q-2023-07-12-1057","pdf_url":"https://quantum-journal.org/papers/q-2023-07-12-1057/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":["crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://quantum-journal.org/papers/q-2023-07-12-1057/pdf/","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5041559873","display_name":"Taylor L. Patti","orcid":"https://orcid.org/0000-0002-4242-6072"},"institutions":[{"id":"https://openalex.org/I136199984","display_name":"Harvard University","ror":"https://ror.org/03vek6s52","country_code":"US","type":"education","lineage":["https://openalex.org/I136199984"]},{"id":"https://openalex.org/I4210127875","display_name":"Nvidia (United States)","ror":"https://ror.org/03jdj4y14","country_code":"US","type":"company","lineage":["https://openalex.org/I4210127875"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"Taylor L. Patti","raw_affiliation_strings":["Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA","NVIDIA, Santa Clara, California 95051, USA"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA","institution_ids":["https://openalex.org/I136199984"]},{"raw_affiliation_string":"NVIDIA, Santa Clara, California 95051, USA","institution_ids":["https://openalex.org/I4210127875"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5006048318","display_name":"Jean Kossaifi","orcid":"https://orcid.org/0000-0002-4445-3429"},"institutions":[{"id":"https://openalex.org/I4210127875","display_name":"Nvidia (United States)","ror":"https://ror.org/03jdj4y14","country_code":"US","type":"company","lineage":["https://openalex.org/I4210127875"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Jean Kossaifi","raw_affiliation_strings":["NVIDIA, Santa Clara, California 95051, USA"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"NVIDIA, Santa Clara, California 95051, USA","institution_ids":["https://openalex.org/I4210127875"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5014498545","display_name":"Anima Anandkumar","orcid":"https://orcid.org/0000-0002-6974-6797"},"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"]},{"id":"https://openalex.org/I4210127875","display_name":"Nvidia (United States)","ror":"https://ror.org/03jdj4y14","country_code":"US","type":"company","lineage":["https://openalex.org/I4210127875"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Anima Anandkumar","raw_affiliation_strings":["Department of Computing + Mathematical Sciences (CMS), California Institute of Technology (Caltech), Pasadena, CA 91125 USA","NVIDIA, Santa Clara, California 95051, USA"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Computing + Mathematical Sciences (CMS), California Institute of Technology (Caltech), Pasadena, CA 91125 USA","institution_ids":["https://openalex.org/I122411786"]},{"raw_affiliation_string":"NVIDIA, Santa Clara, California 95051, USA","institution_ids":["https://openalex.org/I4210127875"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5091317440","display_name":"Susanne F. Yelin","orcid":"https://orcid.org/0000-0003-1655-9151"},"institutions":[{"id":"https://openalex.org/I136199984","display_name":"Harvard University","ror":"https://ror.org/03vek6s52","country_code":"US","type":"education","lineage":["https://openalex.org/I136199984"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Susanne F. Yelin","raw_affiliation_strings":["Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA","institution_ids":["https://openalex.org/I136199984"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":4,"corresponding_author_ids":["https://openalex.org/A5041559873"],"corresponding_institution_ids":["https://openalex.org/I136199984","https://openalex.org/I4210127875"],"apc_list":{"value":200,"currency":"EUR","value_usd":215},"apc_paid":{"value":200,"currency":"EUR","value_usd":215},"fwci":1.8536,"has_fulltext":true,"cited_by_count":11,"citation_normalized_percentile":{"value":0.88220538,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":94,"max":98},"biblio":{"volume":"7","issue":null,"first_page":"1057","last_page":"1057"},"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/T12072","display_name":"Machine Learning and Algorithms","score":0.9979000091552734,"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/T12101","display_name":"Advanced Bandit Algorithms Research","score":0.9977999925613403,"subfield":{"id":"https://openalex.org/subfields/1803","display_name":"Management Science and Operations Research"},"field":{"id":"https://openalex.org/fields/18","display_name":"Decision Sciences"},"domain":{"id":"https://openalex.org/domains/2","display_name":"Social Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.7952338457107544},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.37776798009872437},{"id":"https://openalex.org/keywords/artificial-intelligence","display_name":"Artificial intelligence","score":0.336570143699646},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.32072925567626953}],"concepts":[{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.7952338457107544},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.37776798009872437},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.336570143699646},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.32072925567626953}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.22331/q-2023-07-12-1057","is_oa":true,"landing_page_url":"https://doi.org/10.22331/q-2023-07-12-1057","pdf_url":"https://quantum-journal.org/papers/q-2023-07-12-1057/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:doaj.org/article:5987d1307fd8417b972539cb06bb77ca","is_oa":true,"landing_page_url":"https://doaj.org/article/5987d1307fd8417b972539cb06bb77ca","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 7, p 1057 (2023)","raw_type":"article"}],"best_oa_location":{"id":"doi:10.22331/q-2023-07-12-1057","is_oa":true,"landing_page_url":"https://doi.org/10.22331/q-2023-07-12-1057","pdf_url":"https://quantum-journal.org/papers/q-2023-07-12-1057/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":[],"funders":[{"id":"https://openalex.org/F4320306076","display_name":"National Science Foundation","ror":"https://ror.org/021nxhr62"},{"id":"https://openalex.org/F4320309398","display_name":"California Institute of Technology","ror":"https://ror.org/05dxps055"},{"id":"https://openalex.org/F4320338279","display_name":"Air Force Office of Scientific Research","ror":"https://ror.org/011e9bt93"}],"has_content":{"pdf":true,"grobid_xml":false},"content_urls":{"pdf":"https://content.openalex.org/works/W4384108178.pdf"},"referenced_works_count":70,"referenced_works":["https://openalex.org/W111132822","https://openalex.org/W1524466167","https://openalex.org/W1568345435","https://openalex.org/W1601503375","https://openalex.org/W1669104078","https://openalex.org/W1967781705","https://openalex.org/W1976462214","https://openalex.org/W1983624953","https://openalex.org/W1985123706","https://openalex.org/W1989395545","https://openalex.org/W1996395235","https://openalex.org/W2006051991","https://openalex.org/W2019213021","https://openalex.org/W2065641139","https://openalex.org/W2069525126","https://openalex.org/W2071563913","https://openalex.org/W2085427851","https://openalex.org/W2091560152","https://openalex.org/W2138344304","https://openalex.org/W2149136800","https://openalex.org/W2150865801","https://openalex.org/W2162030326","https://openalex.org/W2346816364","https://openalex.org/W2401610261","https://openalex.org/W2492898698","https://openalex.org/W2564229214","https://openalex.org/W2580787975","https://openalex.org/W2610498683","https://openalex.org/W2735457220","https://openalex.org/W2760965790","https://openalex.org/W2763081248","https://openalex.org/W2768206303","https://openalex.org/W2768413999","https://openalex.org/W2794444783","https://openalex.org/W2797026379","https://openalex.org/W2889426449","https://openalex.org/W2889426946","https://openalex.org/W2948156386","https://openalex.org/W2962726098","https://openalex.org/W2963170238","https://openalex.org/W2964902409","https://openalex.org/W2972228381","https://openalex.org/W2972992123","https://openalex.org/W2995050112","https://openalex.org/W3040484737","https://openalex.org/W3103922448","https://openalex.org/W3104396616","https://openalex.org/W3105463217","https://openalex.org/W3105677655","https://openalex.org/W3111162498","https://openalex.org/W3114970264","https://openalex.org/W3126188815","https://openalex.org/W3133732393","https://openalex.org/W3134798533","https://openalex.org/W3183968998","https://openalex.org/W3199543116","https://openalex.org/W3207697613","https://openalex.org/W3209107739","https://openalex.org/W3214675565","https://openalex.org/W4205585032","https://openalex.org/W4233687788","https://openalex.org/W4245645788","https://openalex.org/W4250589301","https://openalex.org/W4253016148","https://openalex.org/W4255349897","https://openalex.org/W4281647964","https://openalex.org/W4311189852","https://openalex.org/W4313227168","https://openalex.org/W4394671459","https://openalex.org/W6605246359"],"related_works":["https://openalex.org/W2899084033","https://openalex.org/W2748952813","https://openalex.org/W1979597421","https://openalex.org/W2007980826","https://openalex.org/W4245490552","https://openalex.org/W4225152035","https://openalex.org/W2061531152","https://openalex.org/W3002753104","https://openalex.org/W2077600819","https://openalex.org/W2386767533"],"abstract_inverted_index":{"Semidefinite":[0],"programs":[1,66],"are":[2],"optimization":[3,80],"methods":[4,195],"with":[5,67],"a":[6,25,32,47,89,98,118,148,156,197],"wide":[7],"array":[8],"of":[9,50,122,159,168,177,192,200],"applications,":[10],"such":[11,18],"as":[12,88,101,152,154],"approximating":[13],"difficult":[14],"combinatorial":[15],"problems.":[16],"One":[17],"semidefinite":[19,65,139],"program":[20],"is":[21,81],"the":[22,37,85,102,112,123,138,166,178,190,205],"Goemans-Williamson":[23,38,179],"algorithm,":[24],"popular":[26],"integer":[27],"relaxation":[28],"technique.":[29],"We":[30,164],"introduce":[31],"variational":[33],"quantum":[34,175],"algorithm":[35,39,180],"for":[36,181],"that":[40,137],"uses":[41],"only":[42,117],"<mml:math":[43,54,70,74],"xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>n</mml:mi><mml:mrow":[44],"class=\"MJX-TeXAtom-ORD\"><mml:mo>+</mml:mo></mml:mrow><mml:mn>1</mml:mn></mml:math>":[45],"qubits,":[46],"constant":[48],"number":[49,158],"circuit":[51],"preparations,":[52],"and":[53,73],"xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mtext>poly</mml:mtext><mml:mo":[55],"stretchy=\"false\">(</mml:mo><mml:mi>n</mml:mi><mml:mo":[56],"stretchy=\"false\">)</mml:mo></mml:math>":[57,77],"expectation":[58,120,132],"values":[59],"in":[60],"order":[61],"to":[62,69,110],"approximately":[63],"solve":[64],"up":[68],"xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>N</mml:mi><mml:mo>=</mml:mo><mml:msup><mml:mn>2</mml:mn><mml:mi>n</mml:mi></mml:msup></mml:math>":[71],"variables":[72],"xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>M</mml:mi><mml:mo>&amp;#x223C;</mml:mo><mml:mi>O</mml:mi><mml:mo":[75],"stretchy=\"false\">(</mml:mo><mml:mi>N</mml:mi><mml:mo":[76],"constraints.":[78,163],"Efficient":[79],"achieved":[82],"by":[83,115,146,171],"encoding":[84],"objective":[86,113],"matrix":[87],"properly":[90],"parameterized":[91],"unitary":[92],"conditioned":[93],"on":[94,196],"an":[95,173],"auxilary":[96],"qubit,":[97,125],"technique":[99],"known":[100],"Hadamard":[103,106,150],"Test.":[104],"The":[105],"Test":[107],"enables":[108],"us":[109],"optimize":[111],"function":[114],"estimating":[116,129],"single":[119],"value":[121],"ancilla":[124],"rather":[126],"than":[127],"separately":[128],"exponentially":[130],"many":[131],"values.":[133],"Similarly,":[134],"we":[135],"illustrate":[136],"programming":[140],"constraints":[141],"can":[142],"be":[143],"effectively":[144],"enforced":[145],"implementing":[147],"second":[149],"Test,":[151],"well":[153],"imposing":[155],"polynomial":[157],"Pauli":[160],"string":[161],"amplitude":[162],"demonstrate":[165],"effectiveness":[167],"our":[169],"protocol":[170],"devising":[172],"efficient":[174],"implementation":[176],"various":[182],"NP-hard":[183],"problems,":[184],"including":[185],"MaxCut.":[186],"Our":[187],"method":[188],"exceeds":[189],"performance":[191],"analogous":[193],"classical":[194],"diverse":[198],"subset":[199],"well-studied":[201],"MaxCut":[202],"problems":[203],"from":[204],"GSet":[206],"library.":[207]},"counts_by_year":[{"year":2026,"cited_by_count":1},{"year":2025,"cited_by_count":3},{"year":2024,"cited_by_count":5},{"year":2023,"cited_by_count":2}],"updated_date":"2026-05-06T08:25:59.206177","created_date":"2023-07-13T00:00:00"}
