{"id":"https://openalex.org/W3044227471","doi":"https://doi.org/10.1109/tit.2021.3128110","title":"Epsilon-Nets, Unitary Designs, and Random Quantum Circuits","display_name":"Epsilon-Nets, Unitary Designs, and Random Quantum Circuits","publication_year":2021,"publication_date":"2021-11-13","ids":{"openalex":"https://openalex.org/W3044227471","doi":"https://doi.org/10.1109/tit.2021.3128110","mag":"3044227471"},"language":"en","primary_location":{"id":"doi:10.1109/tit.2021.3128110","is_oa":false,"landing_page_url":"https://doi.org/10.1109/tit.2021.3128110","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":["arxiv","crossref","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/2007.10885","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5010835114","display_name":"Micha\u0142 Oszmaniec","orcid":"https://orcid.org/0000-0002-4946-6835"},"institutions":[{"id":"https://openalex.org/I3019243323","display_name":"Center for Theoretical Physics","ror":"https://ror.org/04kfyt897","country_code":"PL","type":"facility","lineage":["https://openalex.org/I3019243323","https://openalex.org/I99542240"]},{"id":"https://openalex.org/I99542240","display_name":"Polish Academy of Sciences","ror":"https://ror.org/01dr6c206","country_code":"PL","type":"government","lineage":["https://openalex.org/I99542240"]}],"countries":["PL"],"is_corresponding":false,"raw_author_name":"Michal Oszmaniec","raw_affiliation_strings":["Center for Theoretical Physics, Polish Academy of Sciences, Warszawa, Poland"],"raw_orcid":"https://orcid.org/0000-0002-4946-6835","affiliations":[{"raw_affiliation_string":"Center for Theoretical Physics, Polish Academy of Sciences, Warszawa, Poland","institution_ids":["https://openalex.org/I3019243323","https://openalex.org/I99542240"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5102974299","display_name":"Adam Sawicki","orcid":"https://orcid.org/0000-0003-4906-2459"},"institutions":[{"id":"https://openalex.org/I3019243323","display_name":"Center for Theoretical Physics","ror":"https://ror.org/04kfyt897","country_code":"PL","type":"facility","lineage":["https://openalex.org/I3019243323","https://openalex.org/I99542240"]},{"id":"https://openalex.org/I99542240","display_name":"Polish Academy of Sciences","ror":"https://ror.org/01dr6c206","country_code":"PL","type":"government","lineage":["https://openalex.org/I99542240"]}],"countries":["PL"],"is_corresponding":false,"raw_author_name":"Adam Sawicki","raw_affiliation_strings":["Center for Theoretical Physics, Polish Academy of Sciences, Warszawa, Poland"],"raw_orcid":"https://orcid.org/0000-0003-4906-2459","affiliations":[{"raw_affiliation_string":"Center for Theoretical Physics, Polish Academy of Sciences, Warszawa, Poland","institution_ids":["https://openalex.org/I3019243323","https://openalex.org/I99542240"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5033198961","display_name":"Micha\u0142 Horodecki","orcid":"https://orcid.org/0000-0002-0446-3059"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Michal Horodecki","raw_affiliation_strings":["International Centre for Theory of Quantum Technologies, University of Gdańsk, Gdańsk, Poland"],"raw_orcid":"https://orcid.org/0000-0002-0446-3059","affiliations":[{"raw_affiliation_string":"International Centre for Theory of Quantum Technologies, University of Gdańsk, Gdańsk, Poland","institution_ids":[]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":0.5887,"has_fulltext":true,"cited_by_count":2,"citation_normalized_percentile":{"value":0.58591976,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":94},"biblio":{"volume":"68","issue":"2","first_page":"989","last_page":"1015"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T12404","display_name":"Mathematical Approximation and Integration","score":0.9998000264167786,"subfield":{"id":"https://openalex.org/subfields/2612","display_name":"Numerical Analysis"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T12404","display_name":"Mathematical Approximation and Integration","score":0.9998000264167786,"subfield":{"id":"https://openalex.org/subfields/2612","display_name":"Numerical Analysis"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11887","display_name":"Quasicrystal Structures and Properties","score":0.9901999831199646,"subfield":{"id":"https://openalex.org/subfields/2505","display_name":"Materials Chemistry"},"field":{"id":"https://openalex.org/fields/25","display_name":"Materials Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T12056","display_name":"Markov Chains and Monte Carlo Methods","score":0.9796000123023987,"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/unitary-state","display_name":"Unitary state","score":0.7162577509880066},{"id":"https://openalex.org/keywords/unitary-matrix","display_name":"Unitary matrix","score":0.6054284572601318},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.5434749722480774},{"id":"https://openalex.org/keywords/dimension","display_name":"Dimension (graph theory)","score":0.5341640114784241},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.4713399112224579},{"id":"https://openalex.org/keywords/quantum-computer","display_name":"Quantum computer","score":0.4609871506690979},{"id":"https://openalex.org/keywords/quantum","display_name":"Quantum","score":0.4304400682449341},{"id":"https://openalex.org/keywords/context","display_name":"Context (archaeology)","score":0.4297890365123749},{"id":"https://openalex.org/keywords/hilbert-space","display_name":"Hilbert space","score":0.42496538162231445},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.396581768989563},{"id":"https://openalex.org/keywords/topology","display_name":"Topology (electrical circuits)","score":0.33414626121520996},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.33364760875701904},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.23835328221321106},{"id":"https://openalex.org/keywords/quantum-mechanics","display_name":"Quantum mechanics","score":0.2148992419242859}],"concepts":[{"id":"https://openalex.org/C67820243","wikidata":"https://www.wikidata.org/wiki/Q179164","display_name":"Unitary state","level":2,"score":0.7162577509880066},{"id":"https://openalex.org/C96214315","wikidata":"https://www.wikidata.org/wiki/Q727103","display_name":"Unitary matrix","level":3,"score":0.6054284572601318},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.5434749722480774},{"id":"https://openalex.org/C33676613","wikidata":"https://www.wikidata.org/wiki/Q13415176","display_name":"Dimension (graph theory)","level":2,"score":0.5341640114784241},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.4713399112224579},{"id":"https://openalex.org/C58053490","wikidata":"https://www.wikidata.org/wiki/Q176555","display_name":"Quantum computer","level":3,"score":0.4609871506690979},{"id":"https://openalex.org/C84114770","wikidata":"https://www.wikidata.org/wiki/Q46344","display_name":"Quantum","level":2,"score":0.4304400682449341},{"id":"https://openalex.org/C2779343474","wikidata":"https://www.wikidata.org/wiki/Q3109175","display_name":"Context (archaeology)","level":2,"score":0.4297890365123749},{"id":"https://openalex.org/C62799726","wikidata":"https://www.wikidata.org/wiki/Q190056","display_name":"Hilbert space","level":2,"score":0.42496538162231445},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.396581768989563},{"id":"https://openalex.org/C184720557","wikidata":"https://www.wikidata.org/wiki/Q7825049","display_name":"Topology (electrical circuits)","level":2,"score":0.33414626121520996},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.33364760875701904},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.23835328221321106},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.2148992419242859},{"id":"https://openalex.org/C151730666","wikidata":"https://www.wikidata.org/wiki/Q7205","display_name":"Paleontology","level":1,"score":0.0},{"id":"https://openalex.org/C17744445","wikidata":"https://www.wikidata.org/wiki/Q36442","display_name":"Political science","level":0,"score":0.0},{"id":"https://openalex.org/C199539241","wikidata":"https://www.wikidata.org/wiki/Q7748","display_name":"Law","level":1,"score":0.0},{"id":"https://openalex.org/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"score":0.0}],"mesh":[],"locations_count":4,"locations":[{"id":"doi:10.1109/tit.2021.3128110","is_oa":false,"landing_page_url":"https://doi.org/10.1109/tit.2021.3128110","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"},{"id":"pmh:oai:arXiv.org:2007.10885","is_oa":true,"landing_page_url":"http://arxiv.org/abs/2007.10885","pdf_url":"https://arxiv.org/pdf/2007.10885","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"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":"","raw_type":null},{"id":"mag:3044227471","is_oa":true,"landing_page_url":"http://arxiv.org/pdf/2007.10885.pdf","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"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":"arXiv (Cornell University)","raw_type":null},{"id":"doi:10.48550/arxiv.2007.10885","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2007.10885","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"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":null,"is_accepted":false,"is_published":null,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2007.10885","is_oa":true,"landing_page_url":"http://arxiv.org/abs/2007.10885","pdf_url":"https://arxiv.org/pdf/2007.10885","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"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":"","raw_type":null},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G1365367362","display_name":null,"funder_award_id":"no.2018/MAB/5","funder_id":"https://openalex.org/F4320321042","funder_display_name":"Fundacja na rzecz Nauki Polskiej"},{"id":"https://openalex.org/G2539852643","display_name":null,"funder_award_id":"04.04.00-00-17C1/18-00","funder_id":"https://openalex.org/F4320321042","funder_display_name":"Fundacja na rzecz Nauki Polskiej"},{"id":"https://openalex.org/G3116650686","display_name":null,"funder_award_id":"POIR.04.04.00-00- 17C1/18-00","funder_id":"https://openalex.org/F4320321042","funder_display_name":"Fundacja na rzecz Nauki Polskiej"},{"id":"https://openalex.org/G381357645","display_name":null,"funder_award_id":"SONATA BIS: 2015/18/E/ST1/00200","funder_id":"https://openalex.org/F4320322650","funder_display_name":"Narodowym Centrum Nauki"},{"id":"https://openalex.org/G7569305816","display_name":null,"funder_award_id":"2018/MAB/5","funder_id":"https://openalex.org/F4320321042","funder_display_name":"Fundacja na rzecz Nauki Polskiej"},{"id":"https://openalex.org/G7830791575","display_name":null,"funder_award_id":"17C1/18-00","funder_id":"https://openalex.org/F4320321042","funder_display_name":"Fundacja na rzecz Nauki Polskiej"}],"funders":[{"id":"https://openalex.org/F4320321042","display_name":"Fundacja na rzecz Nauki Polskiej","ror":"https://ror.org/048zd9m77"},{"id":"https://openalex.org/F4320322650","display_name":"Narodowym Centrum Nauki","ror":"https://ror.org/03ha2q922"}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W3044227471.pdf","grobid_xml":"https://content.openalex.org/works/W3044227471.grobid-xml"},"referenced_works_count":47,"referenced_works":["https://openalex.org/W32507011","https://openalex.org/W1495302720","https://openalex.org/W1502291261","https://openalex.org/W1534413569","https://openalex.org/W1586981851","https://openalex.org/W1595889759","https://openalex.org/W1848366941","https://openalex.org/W1972704456","https://openalex.org/W1991022685","https://openalex.org/W2009751674","https://openalex.org/W2071880083","https://openalex.org/W2084019889","https://openalex.org/W2146720213","https://openalex.org/W2154429330","https://openalex.org/W2290336547","https://openalex.org/W2345171679","https://openalex.org/W2409752923","https://openalex.org/W2482126025","https://openalex.org/W2522375435","https://openalex.org/W2526001183","https://openalex.org/W2528964272","https://openalex.org/W2537703961","https://openalex.org/W2606330852","https://openalex.org/W2612242726","https://openalex.org/W2620384692","https://openalex.org/W2754797405","https://openalex.org/W2756364467","https://openalex.org/W2800865571","https://openalex.org/W2805649731","https://openalex.org/W2953185906","https://openalex.org/W2982169647","https://openalex.org/W2996387336","https://openalex.org/W2999348319","https://openalex.org/W2999578676","https://openalex.org/W3006823398","https://openalex.org/W6632572376","https://openalex.org/W6636412699","https://openalex.org/W6637389729","https://openalex.org/W6667719119","https://openalex.org/W6670820754","https://openalex.org/W6674664011","https://openalex.org/W6695471189","https://openalex.org/W6727355747","https://openalex.org/W6747662415","https://openalex.org/W6754595132","https://openalex.org/W6755357532","https://openalex.org/W6774200370"],"related_works":["https://openalex.org/W3213277701","https://openalex.org/W3081567176","https://openalex.org/W3201561653","https://openalex.org/W3105207217","https://openalex.org/W2944454111","https://openalex.org/W3104711177","https://openalex.org/W1774045736","https://openalex.org/W2962915432","https://openalex.org/W2035288980","https://openalex.org/W2274873607","https://openalex.org/W2947463178","https://openalex.org/W2782664739","https://openalex.org/W2753073040","https://openalex.org/W1559981310","https://openalex.org/W3007969801","https://openalex.org/W2070375169","https://openalex.org/W1944842766","https://openalex.org/W2408048592","https://openalex.org/W1576866980","https://openalex.org/W2896034331"],"abstract_inverted_index":{"Epsilon-nets":[0],"and":[1,20,113,164,204],"approximate":[2,144],"unitary$t$-designs":[3,70],"are":[4],"natural":[5],"notions":[6],"that":[7,60,89,142,143],"capture":[8],"properties":[9],"of":[10,78,92,123,136,157,192,201,223,234,240],"unitary":[11],"operations":[12],"relevant":[13],"for":[14,61,210],"numerous":[15],"applications":[16],"in":[17,133,161,169,231],"quantum":[18,21,137,202,235],"information":[19],"computing.":[22,138],"In":[23],"this":[24],"work":[25],"we":[26,35,87,140,198],"study":[27],"quantitative":[28],"connections":[29],"between":[30],"these":[31],"two":[32],"notions.":[33],"Specifically,":[34],"prove":[36,88,205],"that,":[37],"for$d$dimensional":[38],"Hilbert":[39],"space,":[40],"unitaries":[41],"constituting$\\delta":[42],"$-approximate$t$-expanders":[43],"form$\\epsilon":[44],"$-nets":[45,63],"for$t\\simeq":[46],"\\frac":[47],"{d^{5/2}}{":[48],"\\epsilon":[49,55,73,109],"}$and$\\delta":[50],"\\simeq":[51,72],"\\left":[52],"({\\frac":[53],"{":[54],"^{3/2}}{d}}\\right)^{d^{2}}$.":[56],"We":[57,129],"also":[58],"show":[59,141],"arbitrary$t$,$\\epsilon":[62],"can":[64,146,238],"be":[65,147,181,239],"used":[66],"to":[67,97,180,227],"construct$\\delta":[68],"$-approximate":[69],"for$\\delta":[71],"t$,":[74],"where":[75],"the":[76,83,90,134,162,228,232],"notion":[77],"approximation":[79],"is":[80,219],"based":[81],"on":[82],"diamond":[84],"norm.":[85],"Finally,":[86],"degree":[91],"an":[93],"exact":[94],"unitary$t$design":[95],"necessary":[96],"obtain":[98],"an$\\epsilon":[99],"$-net":[100],"must":[101],"grow":[102],"at":[103],"least":[104],"as":[105],"fast":[106],"as$\\frac":[107],"{1}{":[108],"}$(for":[110],"fixed":[111],"dimension)":[112],"not":[114,179],"slower":[115],"than$d^{2}$(for":[116],"fixed$\\epsilon":[117],"$).":[118],"This":[119],"shows":[120],"near":[121],"optimality":[122],"our":[124,131,175],"result":[125],"connecting$t$-designs":[126],"and$\\epsilon":[127],"$-nets.":[128],"apply":[130],"findings":[132],"context":[135],"First,":[139],"t-designs":[145],"generated":[148],"by":[149],"shallow":[150],"random":[151],"circuits":[152],"formed":[153],"from":[154],"a":[155,206,220],"set":[156],"universal":[158,212],"two-qudit":[159],"gates":[160,185,193,203],"parallel":[163],"sequential":[165],"local":[166],"architectures":[167],"considered":[168],"(Brand\u00e3o":[170],"et":[171],"al.,":[172],"2016).":[173],"Importantly,":[174],"gate":[176,213],"sets":[177],"need":[178],"symmetric":[182],"(i.e.,":[183],"contains":[184],"together":[186],"with":[187,194],"their":[188],"inverses)":[189],"or":[190],"consist":[191],"algebraic":[195],"entries.":[196],"Second,":[197],"consider":[199],"compilation":[200],"non-constructive":[207],"Solovay-Kitaev":[208],"theorem":[209],"general":[211],"sets.":[214],"Our":[215],"main":[216],"technical":[217],"contribution":[218],"new":[221],"construction":[222],"efficient":[224],"polynomial":[225],"approximations":[226],"Dirac":[229],"delta":[230],"space":[233],"channels,":[236],"which":[237],"independent":[241],"interest.":[242]},"counts_by_year":[{"year":2022,"cited_by_count":1},{"year":2021,"cited_by_count":1}],"updated_date":"2026-08-01T09:00:35.917206","created_date":"2025-10-10T00:00:00"}