{"id":"https://openalex.org/W4295004915","doi":"https://doi.org/10.22331/q-2022-09-08-795","title":"Random quantum circuits are approximate unitary <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>t</mml:mi></mml:math>-designs in depth <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>O</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>n</mml:mi><mml:msup><mml:mi>t</mml:mi><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mn>5</mml:mn><mml:mo>+</mml:mo><mml:mi>o</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:msup></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math>","display_name":"Random quantum circuits are approximate unitary <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>t</mml:mi></mml:math>-designs in depth <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>O</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>n</mml:mi><mml:msup><mml:mi>t</mml:mi><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mn>5</mml:mn><mml:mo>+</mml:mo><mml:mi>o</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:msup></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math>","publication_year":2022,"publication_date":"2022-09-08","ids":{"openalex":"https://openalex.org/W4295004915","doi":"https://doi.org/10.22331/q-2022-09-08-795"},"language":"lv","primary_location":{"id":"doi:10.22331/q-2022-09-08-795","is_oa":true,"landing_page_url":"https://doi.org/10.22331/q-2022-09-08-795","pdf_url":"https://quantum-journal.org/papers/q-2022-09-08-795/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-2022-09-08-795/pdf/","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5064304818","display_name":"Jonas Haferkamp","orcid":"https://orcid.org/0000-0002-3592-7274"},"institutions":[{"id":"https://openalex.org/I75951250","display_name":"Freie Universit\u00e4t Berlin","ror":"https://ror.org/046ak2485","country_code":"DE","type":"education","lineage":["https://openalex.org/I75951250"]}],"countries":["DE"],"is_corresponding":true,"raw_author_name":"Jonas Haferkamp","raw_affiliation_strings":["Dahlem Center for Complex Quantum Systems, Freie Universit\u00e4t Berlin, Germany"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Dahlem Center for Complex Quantum Systems, Freie Universit\u00e4t Berlin, Germany","institution_ids":["https://openalex.org/I75951250"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5064304818"],"corresponding_institution_ids":["https://openalex.org/I75951250"],"apc_list":{"value":200,"currency":"EUR","value_usd":215},"apc_paid":{"value":200,"currency":"EUR","value_usd":215},"fwci":9.2976,"has_fulltext":true,"cited_by_count":79,"citation_normalized_percentile":{"value":0.98283458,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":89,"max":100},"biblio":{"volume":"6","issue":null,"first_page":"795","last_page":"795"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11612","display_name":"Stochastic Gradient Optimization Techniques","score":0.9991000294685364,"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/T11612","display_name":"Stochastic Gradient Optimization Techniques","score":0.9991000294685364,"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":0.9980999827384949,"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/T11716","display_name":"Random Matrices and Applications","score":0.9973000288009644,"subfield":{"id":"https://openalex.org/subfields/2613","display_name":"Statistics and Probability"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/quantum-circuit","display_name":"Quantum circuit","score":0.5587864518165588},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.5067086815834045},{"id":"https://openalex.org/keywords/unitary-matrix","display_name":"Unitary matrix","score":0.49780750274658203},{"id":"https://openalex.org/keywords/upper-and-lower-bounds","display_name":"Upper and lower bounds","score":0.47129571437835693},{"id":"https://openalex.org/keywords/quantum-algorithm","display_name":"Quantum algorithm","score":0.44762465357780457},{"id":"https://openalex.org/keywords/quantum","display_name":"Quantum","score":0.43167823553085327},{"id":"https://openalex.org/keywords/randomness","display_name":"Randomness","score":0.4280860722064972},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.42763060331344604},{"id":"https://openalex.org/keywords/haar-measure","display_name":"Haar measure","score":0.42698320746421814},{"id":"https://openalex.org/keywords/pseudorandom-number-generator","display_name":"Pseudorandom number generator","score":0.42341822385787964},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.3125525712966919},{"id":"https://openalex.org/keywords/quantum-mechanics","display_name":"Quantum mechanics","score":0.3117513060569763},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.27833306789398193},{"id":"https://openalex.org/keywords/unitary-state","display_name":"Unitary state","score":0.2727365493774414},{"id":"https://openalex.org/keywords/quantum-error-correction","display_name":"Quantum error correction","score":0.1814393401145935},{"id":"https://openalex.org/keywords/statistics","display_name":"Statistics","score":0.17722493410110474},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.15727216005325317},{"id":"https://openalex.org/keywords/law","display_name":"Law","score":0.0799720287322998}],"concepts":[{"id":"https://openalex.org/C124148022","wikidata":"https://www.wikidata.org/wiki/Q2122210","display_name":"Quantum circuit","level":5,"score":0.5587864518165588},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.5067086815834045},{"id":"https://openalex.org/C96214315","wikidata":"https://www.wikidata.org/wiki/Q727103","display_name":"Unitary matrix","level":3,"score":0.49780750274658203},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.47129571437835693},{"id":"https://openalex.org/C137019171","wikidata":"https://www.wikidata.org/wiki/Q2623817","display_name":"Quantum algorithm","level":3,"score":0.44762465357780457},{"id":"https://openalex.org/C84114770","wikidata":"https://www.wikidata.org/wiki/Q46344","display_name":"Quantum","level":2,"score":0.43167823553085327},{"id":"https://openalex.org/C125112378","wikidata":"https://www.wikidata.org/wiki/Q176640","display_name":"Randomness","level":2,"score":0.4280860722064972},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.42763060331344604},{"id":"https://openalex.org/C130805567","wikidata":"https://www.wikidata.org/wiki/Q1162676","display_name":"Haar measure","level":2,"score":0.42698320746421814},{"id":"https://openalex.org/C140642157","wikidata":"https://www.wikidata.org/wiki/Q1623338","display_name":"Pseudorandom number generator","level":2,"score":0.42341822385787964},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.3125525712966919},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.3117513060569763},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.27833306789398193},{"id":"https://openalex.org/C67820243","wikidata":"https://www.wikidata.org/wiki/Q179164","display_name":"Unitary state","level":2,"score":0.2727365493774414},{"id":"https://openalex.org/C51003876","wikidata":"https://www.wikidata.org/wiki/Q1536431","display_name":"Quantum error correction","level":4,"score":0.1814393401145935},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.17722493410110474},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.15727216005325317},{"id":"https://openalex.org/C199539241","wikidata":"https://www.wikidata.org/wiki/Q7748","display_name":"Law","level":1,"score":0.0799720287322998},{"id":"https://openalex.org/C17744445","wikidata":"https://www.wikidata.org/wiki/Q36442","display_name":"Political science","level":0,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.22331/q-2022-09-08-795","is_oa":true,"landing_page_url":"https://doi.org/10.22331/q-2022-09-08-795","pdf_url":"https://quantum-journal.org/papers/q-2022-09-08-795/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"}],"best_oa_location":{"id":"doi:10.22331/q-2022-09-08-795","is_oa":true,"landing_page_url":"https://doi.org/10.22331/q-2022-09-08-795","pdf_url":"https://quantum-journal.org/papers/q-2022-09-08-795/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/F4320311357","display_name":"Foundational Questions Institute","ror":"https://ror.org/016en1t86"}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W4295004915.pdf","grobid_xml":"https://content.openalex.org/works/W4295004915.grobid-xml"},"referenced_works_count":61,"referenced_works":["https://openalex.org/W1559981310","https://openalex.org/W1848366941","https://openalex.org/W1872835612","https://openalex.org/W1976418263","https://openalex.org/W2007370826","https://openalex.org/W2017477211","https://openalex.org/W2028406173","https://openalex.org/W2039618684","https://openalex.org/W2052146120","https://openalex.org/W2070375169","https://openalex.org/W2089817321","https://openalex.org/W2092385243","https://openalex.org/W2116123253","https://openalex.org/W2129872856","https://openalex.org/W2140422174","https://openalex.org/W2140622323","https://openalex.org/W2146720213","https://openalex.org/W2148118219","https://openalex.org/W2156844769","https://openalex.org/W2170620327","https://openalex.org/W2192515608","https://openalex.org/W2288773634","https://openalex.org/W2290336547","https://openalex.org/W2329539195","https://openalex.org/W2345171679","https://openalex.org/W2405530305","https://openalex.org/W2409432484","https://openalex.org/W2526001183","https://openalex.org/W2759343347","https://openalex.org/W2780472999","https://openalex.org/W2801064005","https://openalex.org/W2804172638","https://openalex.org/W2947463178","https://openalex.org/W2948374647","https://openalex.org/W2950212533","https://openalex.org/W2962915432","https://openalex.org/W2963912534","https://openalex.org/W2996387336","https://openalex.org/W3012928806","https://openalex.org/W3103301792","https://openalex.org/W3104042236","https://openalex.org/W3111056230","https://openalex.org/W3112980360","https://openalex.org/W3123208069","https://openalex.org/W3125368076","https://openalex.org/W3131645664","https://openalex.org/W3138384191","https://openalex.org/W3148666303","https://openalex.org/W3158211506","https://openalex.org/W3172713805","https://openalex.org/W3173924228","https://openalex.org/W3192992028","https://openalex.org/W3213277701","https://openalex.org/W3213820043","https://openalex.org/W4294179124","https://openalex.org/W4297815953","https://openalex.org/W4298350064","https://openalex.org/W4301331119","https://openalex.org/W4321853181","https://openalex.org/W6929427090","https://openalex.org/W6948363148"],"related_works":["https://openalex.org/W3119389895","https://openalex.org/W4287394315","https://openalex.org/W4293583948","https://openalex.org/W2039998427","https://openalex.org/W4302329453","https://openalex.org/W2037476261","https://openalex.org/W2950145299","https://openalex.org/W2184469044","https://openalex.org/W2014037164","https://openalex.org/W3122150010"],"abstract_inverted_index":{"The":[0],"applications":[1,23],"of":[2,17,21,29,76,101,151,184],"random":[3,67,106,156,201,207],"quantum":[4,8,11,30,33,68,107,157,177],"circuits":[5,34,69,108,158],"range":[6],"from":[7],"computing":[9],"and":[10,63,180],"many-body":[12],"systems":[13],"to":[14,26,37,53,122,140,196],"the":[15,27,98,133,181,185,197],"physics":[16],"black":[18],"holes.":[19],"Many":[20],"these":[22],"are":[24,35,45,83],"related":[25],"generation":[28],"pseudorandomness:":[31],"Random":[32],"known":[36],"approximate":[38,84,160],"unitary":[39,85,161],"<mml:math":[40,43,54,78,86,110,123,134,141,144,162,166],"xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>t</mml:mi></mml:math>-designs.":[41,87],"Unitary":[42],"xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>t</mml:mi></mml:math>-designs":[44,163],"probability":[46],"distributions":[47],"that":[48,66,155],"mimic":[49],"Haar":[50,198,206],"randomness":[51],"up":[52],"xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>t</mml:mi></mml:math>th":[55],"moments.":[56],"In":[57,88],"a":[58,73],"seminal":[59],"paper,":[60],"Brand\u00e3o,":[61],"Harrow":[62],"Horodecki":[64],"prove":[65,193],"on":[70],"qubits":[71],"in":[72,164],"brickwork":[74],"architecture":[75],"depth":[77,165],"xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>O</mml:mi><mml:mo":[79,167],"stretchy=\"false\">(</mml:mo><mml:mi>n</mml:mi><mml:msup><mml:mi>t</mml:mi><mml:mrow":[80,168],"class=\"MJX-TeXAtom-ORD\"><mml:mn>10.5</mml:mn></mml:mrow></mml:msup><mml:mo":[81],"stretchy=\"false\">)</mml:mo></mml:math>":[82,137],"this":[89,93,119,152],"work,":[90],"we":[91,192],"revisit":[92],"argument,":[94],"which":[95],"lower":[96,120],"bounds":[97],"spectral":[99],"gap":[100],"moment":[102],"operators":[103],"for":[104,200],"local":[105],"by":[109],"xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi":[111,124],"mathvariant=\"normal\">&amp;#x03A9;</mml:mi><mml:mo":[112,125],"stretchy=\"false\">(</mml:mo><mml:msup><mml:mi>n</mml:mi><mml:mrow":[113,126],"class=\"MJX-TeXAtom-ORD\"><mml:mo>&amp;#x2212;</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:msup><mml:mi>t</mml:mi><mml:mrow":[114,127],"class=\"MJX-TeXAtom-ORD\"><mml:mo>&amp;#x2212;</mml:mo><mml:mn>9.5</mml:mn></mml:mrow></mml:msup><mml:mo":[115],"stretchy=\"false\">)</mml:mo></mml:math>.":[116,172],"We":[117],"improve":[118],"bound":[121,179],"class=\"MJX-TeXAtom-ORD\"><mml:mo>&amp;#x2212;</mml:mo><mml:mn>4</mml:mn><mml:mo>&amp;#x2212;</mml:mo><mml:mi>o</mml:mi><mml:mo":[128],"stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mo":[129,136,170],"stretchy=\"false\">)</mml:mo></mml:mrow></mml:msup><mml:mo":[130,171],"stretchy=\"false\">)</mml:mo></mml:math>,":[131],"where":[132],"xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>o</mml:mi><mml:mo":[135],"term":[138],"goes":[139],"xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mn>0</mml:mn></mml:math>":[142],"as":[143],"xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>t</mml:mi><mml:mo":[145],"stretchy=\"false\">&amp;#x2192;</mml:mo><mml:mi":[146],"mathvariant=\"normal\">&amp;#x221E;</mml:mi></mml:math>.":[147],"A":[148],"direct":[149],"consequence":[150],"scaling":[153],"is":[154],"generate":[159],"class=\"MJX-TeXAtom-ORD\"><mml:mn>5</mml:mn><mml:mo>+</mml:mo><mml:mi>o</mml:mi><mml:mo":[169],"Our":[173],"techniques":[174],"involve":[175],"Gao's":[176],"union":[178],"unreasonable":[182],"effectiveness":[183],"Clifford":[186,202],"group.":[187],"As":[188],"an":[189],"auxiliary":[190],"result,":[191],"fast":[194],"convergence":[195],"measure":[199],"unitaries":[203],"interleaved":[204],"with":[205],"single":[208],"qubit":[209],"unitaries.":[210]},"counts_by_year":[{"year":2026,"cited_by_count":11},{"year":2025,"cited_by_count":31},{"year":2024,"cited_by_count":18},{"year":2023,"cited_by_count":16},{"year":2022,"cited_by_count":2},{"year":2021,"cited_by_count":1}],"updated_date":"2026-06-05T09:01:59.212387","created_date":"2022-09-09T00:00:00"}
