{"id":"https://openalex.org/W1979308021","doi":"https://doi.org/10.1137/s0097539705447177","title":"The Power of Strong Fourier Sampling: Quantum Algorithms for Affine Groups and Hidden Shifts","display_name":"The Power of Strong Fourier Sampling: Quantum Algorithms for Affine Groups and Hidden Shifts","publication_year":2007,"publication_date":"2007-01-01","ids":{"openalex":"https://openalex.org/W1979308021","doi":"https://doi.org/10.1137/s0097539705447177","mag":"1979308021"},"language":"en","primary_location":{"id":"doi:10.1137/s0097539705447177","is_oa":false,"landing_page_url":"https://doi.org/10.1137/s0097539705447177","pdf_url":null,"source":{"id":"https://openalex.org/S153560523","display_name":"SIAM Journal on Computing","issn_l":"0097-5397","issn":["0097-5397","1095-7111"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Computing","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/A5008033989","display_name":"Cristopher Moore","orcid":"https://orcid.org/0000-0002-2062-1942"},"institutions":[{"id":"https://openalex.org/I1308548392","display_name":"Santa Fe Institute","ror":"https://ror.org/01arysc35","country_code":"US","type":"funder","lineage":["https://openalex.org/I1308548392"]},{"id":"https://openalex.org/I169521973","display_name":"University of New Mexico","ror":"https://ror.org/05fs6jp91","country_code":"US","type":"education","lineage":["https://openalex.org/I169521973"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Cristopher Moore","raw_affiliation_strings":["Santa Fe Institute, Santa Fe, NM 87501","University of New Mexico, Albuquerque, NM 87131, and"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Santa Fe Institute, Santa Fe, NM 87501","institution_ids":["https://openalex.org/I1308548392"]},{"raw_affiliation_string":"University of New Mexico, Albuquerque, NM 87131, and","institution_ids":["https://openalex.org/I169521973"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5072455364","display_name":"Daniel N. Rockmore","orcid":"https://orcid.org/0000-0001-5411-9224"},"institutions":[{"id":"https://openalex.org/I107672454","display_name":"Dartmouth College","ror":"https://ror.org/049s0rh22","country_code":"US","type":"education","lineage":["https://openalex.org/I107672454"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Daniel Rockmore","raw_affiliation_strings":["Dartmouth College, Hanover, NH 03755"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Dartmouth College, Hanover, NH 03755","institution_ids":["https://openalex.org/I107672454"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5088846595","display_name":"Alexander Russell","orcid":"https://orcid.org/0000-0002-8228-6238"},"institutions":[{"id":"https://openalex.org/I140172145","display_name":"University of Connecticut","ror":"https://ror.org/02der9h97","country_code":"US","type":"education","lineage":["https://openalex.org/I140172145"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Alexander Russell","raw_affiliation_strings":["University of Connecticut, Storrs, CT 06269"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"University of Connecticut, Storrs, CT 06269","institution_ids":["https://openalex.org/I140172145"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5065859226","display_name":"Leonard J. Schulman","orcid":"https://orcid.org/0000-0001-9901-2797"},"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":"Leonard J. Schulman","raw_affiliation_strings":["California Institute of Technology, Pasadena, CA 91125"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"California Institute of Technology, Pasadena, CA 91125","institution_ids":["https://openalex.org/I122411786"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":5,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":3.3233,"has_fulltext":false,"cited_by_count":46,"citation_normalized_percentile":{"value":0.92469416,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":89,"max":98},"biblio":{"volume":"37","issue":"3","first_page":"938","last_page":"958"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10682","display_name":"Quantum Computing Algorithms and Architecture","score":1.0,"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":1.0,"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":0.9987000226974487,"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":0.9940999746322632,"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/quantum-fourier-transform","display_name":"Quantum Fourier transform","score":0.7303069829940796},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6805303692817688},{"id":"https://openalex.org/keywords/coset","display_name":"Coset","score":0.6652820706367493},{"id":"https://openalex.org/keywords/quantum-algorithm","display_name":"Quantum algorithm","score":0.6223187446594238},{"id":"https://openalex.org/keywords/fourier-transform","display_name":"Fourier transform","score":0.557875394821167},{"id":"https://openalex.org/keywords/discrete-fourier-transform","display_name":"Discrete Fourier transform (general)","score":0.5289860963821411},{"id":"https://openalex.org/keywords/quantum","display_name":"Quantum","score":0.4586581587791443},{"id":"https://openalex.org/keywords/polynomial","display_name":"Polynomial","score":0.4387814402580261},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.42482930421829224},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.4232429265975952},{"id":"https://openalex.org/keywords/affine-transformation","display_name":"Affine transformation","score":0.4142116606235504},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.40787985920906067},{"id":"https://openalex.org/keywords/fourier-analysis","display_name":"Fourier analysis","score":0.2782110571861267},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.27433353662490845},{"id":"https://openalex.org/keywords/fractional-fourier-transform","display_name":"Fractional Fourier transform","score":0.25660181045532227},{"id":"https://openalex.org/keywords/quantum-mechanics","display_name":"Quantum mechanics","score":0.17373624444007874},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.16922950744628906},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.15238061547279358},{"id":"https://openalex.org/keywords/quantum-gate","display_name":"Quantum gate","score":0.09215912222862244}],"concepts":[{"id":"https://openalex.org/C59500034","wikidata":"https://www.wikidata.org/wiki/Q1464944","display_name":"Quantum Fourier transform","level":5,"score":0.7303069829940796},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6805303692817688},{"id":"https://openalex.org/C85307737","wikidata":"https://www.wikidata.org/wiki/Q751969","display_name":"Coset","level":2,"score":0.6652820706367493},{"id":"https://openalex.org/C137019171","wikidata":"https://www.wikidata.org/wiki/Q2623817","display_name":"Quantum algorithm","level":3,"score":0.6223187446594238},{"id":"https://openalex.org/C102519508","wikidata":"https://www.wikidata.org/wiki/Q6520159","display_name":"Fourier transform","level":2,"score":0.557875394821167},{"id":"https://openalex.org/C57733114","wikidata":"https://www.wikidata.org/wiki/Q1006032","display_name":"Discrete Fourier transform (general)","level":5,"score":0.5289860963821411},{"id":"https://openalex.org/C84114770","wikidata":"https://www.wikidata.org/wiki/Q46344","display_name":"Quantum","level":2,"score":0.4586581587791443},{"id":"https://openalex.org/C90119067","wikidata":"https://www.wikidata.org/wiki/Q43260","display_name":"Polynomial","level":2,"score":0.4387814402580261},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.42482930421829224},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.4232429265975952},{"id":"https://openalex.org/C92757383","wikidata":"https://www.wikidata.org/wiki/Q382497","display_name":"Affine transformation","level":2,"score":0.4142116606235504},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.40787985920906067},{"id":"https://openalex.org/C203024314","wikidata":"https://www.wikidata.org/wiki/Q1365258","display_name":"Fourier analysis","level":3,"score":0.2782110571861267},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.27433353662490845},{"id":"https://openalex.org/C76563020","wikidata":"https://www.wikidata.org/wiki/Q4817582","display_name":"Fractional Fourier transform","level":4,"score":0.25660181045532227},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.17373624444007874},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.16922950744628906},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.15238061547279358},{"id":"https://openalex.org/C58849907","wikidata":"https://www.wikidata.org/wiki/Q2118982","display_name":"Quantum gate","level":4,"score":0.09215912222862244}],"mesh":[],"locations_count":4,"locations":[{"id":"doi:10.1137/s0097539705447177","is_oa":false,"landing_page_url":"https://doi.org/10.1137/s0097539705447177","pdf_url":null,"source":{"id":"https://openalex.org/S153560523","display_name":"SIAM Journal on Computing","issn_l":"0097-5397","issn":["0097-5397","1095-7111"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Computing","raw_type":"journal-article"},{"id":"pmh:oai:authors.library.caltech.edu:9092","is_oa":false,"landing_page_url":"https://authors.library.caltech.edu/9092/","pdf_url":null,"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":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"Article"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.155.4717","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.155.4717","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://authors.library.caltech.edu/9092/1/MOOsiamjc07.pdf","raw_type":"text"},{"id":"pmh:oai:cds.cern.ch:826696","is_oa":false,"landing_page_url":"http://cds.cern.ch/record/826696","pdf_url":null,"source":{"id":"https://openalex.org/S4306402195","display_name":"CERN Document Server (European Organization for Nuclear Research)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I67311998","host_organization_name":"European Organization for Nuclear Research","host_organization_lineage":["https://openalex.org/I67311998"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":null}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G1239072055","display_name":null,"funder_award_id":"CCR-0093065","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G3873147427","display_name":"ITR Collaborative Research:  Complexity-Theoretic Applications of Fourier Analysis","funder_award_id":"0220070","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G4100315099","display_name":"CAREER: Efficient Cryptography with Provable Security Guarantees","funder_award_id":"0093065","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G4643048668","display_name":"ITR Collaborative Research:  Complexity-Theoretic Applications of Fourier Analysis","funder_award_id":"0220264","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G6901105303","display_name":"QnTM: Collaborative Research: The Quantum Complexity of Algebraic Problems","funder_award_id":"0524613","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G7615268006","display_name":"Collaborative Research: Quantum Monte Carlo Algorithms and Quantum Circuit Complexity","funder_award_id":"0218563","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G776827577","display_name":"Collaborative Research: Quantum Monte Carlo Algorithms and quantum circuit complexity","funder_award_id":"0218443","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G869850891","display_name":"CAREER: Computation Methods","funder_award_id":"0049092","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/F4320309398","display_name":"California Institute of Technology","ror":"https://ror.org/05dxps055"},{"id":"https://openalex.org/F4320310821","display_name":"Charles Lee Powell Foundation","ror":"https://ror.org/02h144281"},{"id":"https://openalex.org/F4320316838","display_name":"Kempe Foundation","ror":null},{"id":"https://openalex.org/F4320337380","display_name":"Division of Mathematical Sciences","ror":"https://ror.org/051fftw81"}],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":25,"referenced_works":["https://openalex.org/W1534413569","https://openalex.org/W1541824250","https://openalex.org/W1559738570","https://openalex.org/W1596185498","https://openalex.org/W1632987656","https://openalex.org/W1640526923","https://openalex.org/W1896491785","https://openalex.org/W1967900748","https://openalex.org/W1978883801","https://openalex.org/W1995766457","https://openalex.org/W2015055353","https://openalex.org/W2025367213","https://openalex.org/W2032830293","https://openalex.org/W2035742807","https://openalex.org/W2044747430","https://openalex.org/W2055201875","https://openalex.org/W2057065544","https://openalex.org/W2097358487","https://openalex.org/W2117584890","https://openalex.org/W2140496956","https://openalex.org/W2156190205","https://openalex.org/W2595131947","https://openalex.org/W2784965142","https://openalex.org/W3023478445","https://openalex.org/W3111297213"],"related_works":["https://openalex.org/W3136237069","https://openalex.org/W2801113937","https://openalex.org/W2953274551","https://openalex.org/W2065276051","https://openalex.org/W2952898040","https://openalex.org/W1978883801","https://openalex.org/W4224937327","https://openalex.org/W2138874081","https://openalex.org/W2891491794","https://openalex.org/W4311689615"],"abstract_inverted_index":{"Many":[0],"quantum":[1,34,60,226],"algorithms,":[2,10],"including":[3],"Shor's":[4],"celebrated":[5],"factoring":[6],"and":[7,67,96,107,192,227,248,285,330],"discrete":[8],"log":[9],"proceed":[11],"by":[12,56,152,340],"reduction":[13],"to":[14,234,277,290,314],"a":[15,26,33,44,113,222,251,342],"hidden":[16,50,140,173,311,320,354],"subgroup":[17,23,51,303,312,355],"problem,":[18],"in":[19,112,167,193,242,250,295],"which":[20,352],"an":[21,296],"unknown":[22],"H":[24,175],"of":[25,47,63,78,109,176,225,257,272,282,317,326,334,349],"group":[27,72],"G":[28,38],"must":[29],"be":[30,144,200,218,358],"determined":[31],"from":[32],"state":[35],"$\\psi$":[36,64],"over":[37,351],"that":[39,135,142,172,238,255,299],"is":[40,65,73,132,275],"uniformly":[41],"supported":[42],"on":[43,332],"left":[45],"coset":[46],"H.":[48],"These":[49],"problems":[52],"are":[53,94,139,259],"typically":[54],"solved":[55,359],"Fourier":[57,61,80,287],"sampling:":[58],"the":[59,70,79,86,97,105,110,128,147,153,168,177,195,204,243,262,279,292,301,310,324,347,353],"transform":[62],"computed":[66],"measured.":[68],"When":[69],"underlying":[71],"nonabelian,":[74],"two":[75,235],"important":[76],"variants":[77],"sampling":[81],"paradigm":[82],"have":[83,239],"been":[84,240],"identified:":[85],"weak":[87],"standard":[88,99,130,206,264],"method,":[89,100,247],"where":[90,101,188],"only":[91],"representation":[92,283],"names":[93],"measured,":[95],"strong":[98,129,148,205,263],"full":[102,280],"measurement":[103,249],"(i.e.,":[104],"row":[106],"column":[108],"representation,":[111],"suitably":[114],"chosen":[115],"basis,":[116],"as":[117,119],"well":[118],"its":[120],"name)":[121],"occurs.":[122],"It":[123],"has":[124],"remained":[125],"open":[126],"whether":[127,137],"method":[131,149],"indeed":[133],"stronger,":[134],"is,":[136],"there":[138],"subgroups":[141,174,216],"can":[143,199,217,357],"reconstructed":[145,202,220],"via":[146],"but":[150],"not":[151],"weak,":[154],"or":[155],"any":[156],"other":[157],"known,":[158],"method.":[159,207,265],"In":[160],"this":[161,165],"article,":[162],"we":[163,338],"settle":[164],"question":[166],"affirmative.":[169],"We":[170,230,305],"show":[171,254],"q-hedral":[178],"groups,":[179,273],"i.e.,":[180],"semidirect":[181],"products":[182],"${\\mathbb":[183],"Z}_q":[184],"\\ltimes":[185],"{\\mathbb":[186],"Z}_p$,":[187],"$q":[189],"\\mid":[190],"(p-1)$,":[191],"particular":[194],"affine":[196],"groups":[197,350],"$A_p$,":[198],"information-theoretically":[201],"using":[203],"Moreover,":[208],"if":[209],"$|H|":[210],"=":[211],"p/":[212],"{\\rm":[213],"polylog}(p)$,":[214],"these":[215,258],"fully":[219],"with":[221],"polynomial":[223],"amount":[224],"classical":[228],"computation.":[229],"compare":[231],"our":[232,307],"algorithms":[233],"weaker":[236,260],"methods":[237],"discussed":[241],"literature\u2014the":[244],"\u201cforgetful\u201d":[245],"abelian":[246],"random":[252],"basis\u2014and":[253],"both":[256],"than":[261],"Thus,":[266],"at":[267],"least":[268],"for":[269,309,346],"some":[270],"families":[271,316],"it":[274],"crucial":[276],"use":[278],"power":[281],"theory":[284],"nonabelian":[286],"analysis,":[288],"namely,":[289],"measure":[291],"high-dimensional":[293],"representations":[294],"adapted":[297],"basis":[298],"respects":[300],"group's":[302],"structure.":[304],"apply":[306],"algorithm":[308],"problem":[313,356],"new":[315],"cryptographically":[318],"motivated":[319],"shift":[321],"problems,":[322],"generalizing":[323],"work":[325],"van":[327],"Dam,":[328],"Hallgren,":[329],"Ip":[331],"shifts":[333],"multiplicative":[335],"characters.":[336],"Finally,":[337],"close":[339],"proving":[341],"simple":[343],"closure":[344],"property":[345],"class":[348],"efficiently.":[360]},"counts_by_year":[{"year":2022,"cited_by_count":3},{"year":2017,"cited_by_count":1},{"year":2016,"cited_by_count":2},{"year":2015,"cited_by_count":1},{"year":2014,"cited_by_count":5},{"year":2013,"cited_by_count":2},{"year":2012,"cited_by_count":1}],"updated_date":"2026-07-28T07:46:37.118299","created_date":"2025-10-10T00:00:00"}