{"id":"https://openalex.org/W3168912476","doi":"https://doi.org/10.22331/q-2022-02-15-652","title":"Lower Bounds on Stabilizer Rank","display_name":"Lower Bounds on Stabilizer Rank","publication_year":2022,"publication_date":"2022-02-15","ids":{"openalex":"https://openalex.org/W3168912476","doi":"https://doi.org/10.22331/q-2022-02-15-652","mag":"3168912476"},"language":"lv","primary_location":{"id":"doi:10.22331/q-2022-02-15-652","is_oa":true,"landing_page_url":"https://doi.org/10.22331/q-2022-02-15-652","pdf_url":"https://quantum-journal.org/papers/q-2022-02-15-652/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":"preprint","indexed_in":["arxiv","crossref","datacite","doaj"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://quantum-journal.org/papers/q-2022-02-15-652/pdf/","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5070527290","display_name":"Shir Peleg","orcid":"https://orcid.org/0000-0002-7836-7780"},"institutions":[{"id":"https://openalex.org/I16391192","display_name":"Tel Aviv University","ror":"https://ror.org/04mhzgx49","country_code":"IL","type":"education","lineage":["https://openalex.org/I16391192"]}],"countries":["IL"],"is_corresponding":false,"raw_author_name":"Shir Peleg","raw_affiliation_strings":["Blavatnik School of Computer Science, Tel Aviv University, Tel Aviv, Israel","Tel Aviv University *"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Blavatnik School of Computer Science, Tel Aviv University, Tel Aviv, Israel","institution_ids":["https://openalex.org/I16391192"]},{"raw_affiliation_string":"Tel Aviv University *","institution_ids":["https://openalex.org/I16391192"]}]},{"author_position":"middle","author":{"id":null,"display_name":"Amir Shpilka","orcid":null},"institutions":[{"id":"https://openalex.org/I16391192","display_name":"Tel Aviv University","ror":"https://ror.org/04mhzgx49","country_code":"IL","type":"education","lineage":["https://openalex.org/I16391192"]}],"countries":["IL"],"is_corresponding":false,"raw_author_name":"Amir Shpilka","raw_affiliation_strings":["Blavatnik School of Computer Science, Tel Aviv University, Tel Aviv, Israel","Tel Aviv University *"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Blavatnik School of Computer Science, Tel Aviv University, Tel Aviv, Israel","institution_ids":["https://openalex.org/I16391192"]},{"raw_affiliation_string":"Tel Aviv University *","institution_ids":["https://openalex.org/I16391192"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5073175433","display_name":"Ben Lee Volk","orcid":"https://orcid.org/0000-0002-7143-7280"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Ben Lee Volk","raw_affiliation_strings":["Efi Arazi School of Computer Science, Reichman University, Herzliya, Israel"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Efi Arazi School of Computer Science, Reichman University, Herzliya, Israel","institution_ids":[]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":3,"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":0.1387,"has_fulltext":true,"cited_by_count":1,"citation_normalized_percentile":{"value":0.50905474,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":94},"biblio":{"volume":"6","issue":null,"first_page":"652","last_page":"652"},"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/T10720","display_name":"Complexity and Algorithms in Graphs","score":0.9976000189781189,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"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/T10363","display_name":"Low-power high-performance VLSI design","score":0.9943000078201294,"subfield":{"id":"https://openalex.org/subfields/2208","display_name":"Electrical and Electronic Engineering"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/rank","display_name":"Rank (graph theory)","score":0.6640737056732178},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.6349527835845947},{"id":"https://openalex.org/keywords/upper-and-lower-bounds","display_name":"Upper and lower bounds","score":0.5563659071922302},{"id":"https://openalex.org/keywords/omega","display_name":"Omega","score":0.5371608734130859},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.521898090839386},{"id":"https://openalex.org/keywords/state","display_name":"State (computer science)","score":0.5128557682037354},{"id":"https://openalex.org/keywords/stabilizer","display_name":"Stabilizer (aeronautics)","score":0.4831520915031433},{"id":"https://openalex.org/keywords/quadratic-equation","display_name":"Quadratic equation","score":0.4806303381919861},{"id":"https://openalex.org/keywords/degree","display_name":"Degree (music)","score":0.43707743287086487},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.38969284296035767},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.28866565227508545},{"id":"https://openalex.org/keywords/quantum-mechanics","display_name":"Quantum mechanics","score":0.2585263252258301},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.08815032243728638}],"concepts":[{"id":"https://openalex.org/C164226766","wikidata":"https://www.wikidata.org/wiki/Q7293202","display_name":"Rank (graph theory)","level":2,"score":0.6640737056732178},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.6349527835845947},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.5563659071922302},{"id":"https://openalex.org/C2779557605","wikidata":"https://www.wikidata.org/wiki/Q9890","display_name":"Omega","level":2,"score":0.5371608734130859},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.521898090839386},{"id":"https://openalex.org/C48103436","wikidata":"https://www.wikidata.org/wiki/Q599031","display_name":"State (computer science)","level":2,"score":0.5128557682037354},{"id":"https://openalex.org/C2777502316","wikidata":"https://www.wikidata.org/wiki/Q2712515","display_name":"Stabilizer (aeronautics)","level":2,"score":0.4831520915031433},{"id":"https://openalex.org/C129844170","wikidata":"https://www.wikidata.org/wiki/Q41299","display_name":"Quadratic equation","level":2,"score":0.4806303381919861},{"id":"https://openalex.org/C2775997480","wikidata":"https://www.wikidata.org/wiki/Q586277","display_name":"Degree (music)","level":2,"score":0.43707743287086487},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.38969284296035767},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.28866565227508545},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.2585263252258301},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.08815032243728638},{"id":"https://openalex.org/C78519656","wikidata":"https://www.wikidata.org/wiki/Q101333","display_name":"Mechanical engineering","level":1,"score":0.0},{"id":"https://openalex.org/C127413603","wikidata":"https://www.wikidata.org/wiki/Q11023","display_name":"Engineering","level":0,"score":0.0},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.0},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.0},{"id":"https://openalex.org/C24890656","wikidata":"https://www.wikidata.org/wiki/Q82811","display_name":"Acoustics","level":1,"score":0.0}],"mesh":[],"locations_count":8,"locations":[{"id":"doi:10.22331/q-2022-02-15-652","is_oa":true,"landing_page_url":"https://doi.org/10.22331/q-2022-02-15-652","pdf_url":"https://quantum-journal.org/papers/q-2022-02-15-652/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:2106.03214","is_oa":true,"landing_page_url":"http://arxiv.org/abs/2106.03214","pdf_url":"https://arxiv.org/pdf/2106.03214","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":"mag:3168912476","is_oa":true,"landing_page_url":"https://arxiv.org/pdf/2106.03214","pdf_url":null,"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":"arXiv (Cornell University)","raw_type":null},{"id":"pmh:oai:doaj.org/article:3a9558f492d040f8b6753a2b19bdf61c","is_oa":true,"landing_page_url":"https://doaj.org/article/3a9558f492d040f8b6753a2b19bdf61c","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 652 (2022)","raw_type":"article"},{"id":"pmh:oai:drops-oai.dagstuhl.de:15706","is_oa":true,"landing_page_url":"https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2022.110","pdf_url":null,"source":{"id":"https://openalex.org/S4377196569","display_name":"DROPS (Schloss Dagstuhl \u2013 Leibniz Center for Informatics)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I2799853480","host_organization_name":"Schloss Dagstuhl \u2013 Leibniz Center for Informatics","host_organization_lineage":["https://openalex.org/I2799853480"],"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":"InProceedings"},{"id":"doi:10.48550/arxiv.2106.03214","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2106.03214","pdf_url":null,"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":null,"is_accepted":false,"is_published":null,"raw_source_name":null,"raw_type":"article-journal"},{"id":"mag:3176763179","is_oa":true,"landing_page_url":"https://arxiv.org/abs/2106.03214","pdf_url":null,"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":null,"is_accepted":false,"is_published":null,"raw_source_name":"arXiv (Cornell University)","raw_type":null},{"id":"doi:10.4230/lipics.itcs.2022.110","is_oa":true,"landing_page_url":"https://doi.org/10.4230/lipics.itcs.2022.110","pdf_url":null,"source":{"id":"https://openalex.org/S7407052059","display_name":"Dagstuhl Research Online Publication Server","issn_l":null,"issn":[],"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","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":null,"raw_source_name":null,"raw_type":""}],"best_oa_location":{"id":"doi:10.22331/q-2022-02-15-652","is_oa":true,"landing_page_url":"https://doi.org/10.22331/q-2022-02-15-652","pdf_url":"https://quantum-journal.org/papers/q-2022-02-15-652/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/G1723858993","display_name":null,"funder_award_id":"CCF-1705028","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G3049538294","display_name":null,"funder_award_id":"514/20","funder_id":"https://openalex.org/F4320322252","funder_display_name":"Israel Science Foundation"},{"id":"https://openalex.org/G402810861","display_name":null,"funder_award_id":"1705028","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/F4320312914","display_name":"Blavatnik Family Foundation","ror":null},{"id":"https://openalex.org/F4320322252","display_name":"Israel Science Foundation","ror":"https://ror.org/04sazxf24"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W3168912476.pdf","grobid_xml":"https://content.openalex.org/works/W3168912476.grobid-xml"},"referenced_works_count":32,"referenced_works":["https://openalex.org/W32507011","https://openalex.org/W1789015869","https://openalex.org/W1839785972","https://openalex.org/W2009169733","https://openalex.org/W2009994177","https://openalex.org/W2033216491","https://openalex.org/W2047370658","https://openalex.org/W2052146120","https://openalex.org/W2070678975","https://openalex.org/W2079294297","https://openalex.org/W2084652510","https://openalex.org/W2117584890","https://openalex.org/W2142610181","https://openalex.org/W2162699574","https://openalex.org/W2181445345","https://openalex.org/W2299781645","https://openalex.org/W2611715903","https://openalex.org/W2789121304","https://openalex.org/W2885845431","https://openalex.org/W2949864545","https://openalex.org/W2952315951","https://openalex.org/W2963744259","https://openalex.org/W2982509980","https://openalex.org/W2990961515","https://openalex.org/W3023478445","https://openalex.org/W3037871222","https://openalex.org/W3105782957","https://openalex.org/W3106326670","https://openalex.org/W3111297213","https://openalex.org/W3170369836","https://openalex.org/W4210907418","https://openalex.org/W6958769664"],"related_works":["https://openalex.org/W3176763179","https://openalex.org/W3083750962","https://openalex.org/W2952133313","https://openalex.org/W2112553910","https://openalex.org/W2092016058","https://openalex.org/W1597745083","https://openalex.org/W3058516402","https://openalex.org/W2806100743","https://openalex.org/W2964024102","https://openalex.org/W3197412015","https://openalex.org/W1997575430","https://openalex.org/W2949394968","https://openalex.org/W1508725836","https://openalex.org/W3094151347","https://openalex.org/W1883887292","https://openalex.org/W2770274520","https://openalex.org/W2610386774","https://openalex.org/W2283308379","https://openalex.org/W2982420825","https://openalex.org/W2128662225"],"abstract_inverted_index":{"The":[0,33,167],"<mml:math":[1,11,16,20,24,31,52,65,83,103,113,119,151],"xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mrow":[2],"class=\"MJX-TeXAtom-ORD\"><mml:mtext":[3],"class=\"MJX-tex-mathit\"":[4],"mathvariant=\"italic\">stabilizer":[5],"rank</mml:mtext></mml:mrow></mml:math>":[6],"of":[7,36,50,56,64,74,82,88,108,141,150,161,169,177,180,186],"a":[8,61,78,99,174],"quantum":[9,42],"state":[10,110],"xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>&amp;#x03C8;</mml:mi></mml:math>":[12],"is":[13,44,112,118,126],"the":[14,47,51,71,105,127,139,170,184,187,200],"minimal":[15],"xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>r</mml:mi></mml:math>":[17],"such":[18,75],"that":[19,97],"xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mrow><mml:mo>|</mml:mo><mml:mi>&amp;#x03C8;</mml:mi><mml:mo>&amp;#x27E9;</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:munderover><mml:mo>&amp;#x2211;</mml:mo><mml:mrow":[21],"class=\"MJX-TeXAtom-ORD\"><mml:mi>j</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mi>r</mml:mi></mml:munderover><mml:msub><mml:mi>c</mml:mi><mml:mi>j</mml:mi></mml:msub><mml:mrow><mml:mo>|</mml:mo><mml:msub><mml:mi>&amp;#x03C6;</mml:mi><mml:mi>j</mml:mi></mml:msub><mml:mo>&amp;#x27E9;</mml:mo></mml:mrow></mml:math>":[22],"for":[23,41,98,132],"xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:msub><mml:mi>c</mml:mi><mml:mi>j</mml:mi></mml:msub><mml:mo>&amp;#x2208;</mml:mo><mml:mrow":[25],"class=\"MJX-TeXAtom-ORD\"><mml:mi":[26,153],"mathvariant=\"double-struck\">C</mml:mi></mml:mrow></mml:math>":[27],"and":[28,91,155,164,196],"stabilizer":[29,48,72,106,134,142],"states":[30,117,143],"xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:msub><mml:mi>&amp;#x03C6;</mml:mi><mml:mi>j</mml:mi></mml:msub></mml:math>.":[32],"running":[34],"time":[35],"several":[37],"classical":[38],"simulation":[39],"methods":[40],"circuits":[43],"determined":[45],"by":[46],"rank":[49,73,107],"xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>n</mml:mi></mml:math>-th":[53],"tensor":[54],"power":[55],"single-qubit":[57],"magic":[58],"states.We":[59],"prove":[60,96],"lower":[62,80,130],"bound":[63,81,131],"xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi":[66,84,120],"mathvariant=\"normal\">&amp;#x03A9;</mml:mi><mml:mo":[67,85,121],"stretchy=\"false\">(</mml:mo><mml:mi>n</mml:mi><mml:mo":[68],"stretchy=\"false\">)</mml:mo></mml:math>":[69,87],"on":[70,138],"states,":[76],"improving":[77],"previous":[79],"stretchy=\"false\">(</mml:mo><mml:msqrt><mml:mi>n</mml:mi></mml:msqrt><mml:mo":[86],"Bravyi,":[89],"Smith":[90],"Smolin":[92],"\\cite{BSS16}.":[93],"Further,":[94],"we":[95,156],"sufficiently":[100],"small":[101],"constant":[102],"xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>&amp;#x03B4;</mml:mi></mml:math>,":[104],"any":[109],"which":[111],"xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>&amp;#x03B4;</mml:mi></mml:math>-close":[114],"to":[115],"those":[116],"stretchy=\"false\">(</mml:mo><mml:msqrt><mml:mi>n</mml:mi></mml:msqrt><mml:mrow":[122],"class=\"MJX-TeXAtom-ORD\"><mml:mo>/</mml:mo></mml:mrow><mml:mi>log</mml:mi><mml:mo>&amp;#x2061;</mml:mo><mml:mi>n</mml:mi><mml:mo":[123],"stretchy=\"false\">)</mml:mo></mml:math>.":[124],"This":[125],"first":[128,171],"non-trivial":[129],"approximate":[133],"rank.Our":[135],"techniques":[136],"rely":[137],"representation":[140],"as":[144],"quadratic":[145,181],"functions":[146,163],"over":[147],"affine":[148],"subspaces":[149],"xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:msubsup><mml:mrow":[152],"mathvariant=\"double-struck\">F</mml:mi></mml:mrow><mml:mn>2</mml:mn><mml:mi>n</mml:mi></mml:msubsup></mml:math>,":[154],"use":[157],"tools":[158],"from":[159],"analysis":[160,176],"boolean":[162],"complexity":[165],"theory.":[166],"proof":[168,185],"result":[172,189],"involves":[173],"careful":[175],"directional":[178],"derivatives":[179],"polynomials,":[182],"whereas":[183],"second":[188],"uses":[190],"Razborov-Smolensky":[191],"low":[192],"degree":[193],"polynomial":[194],"approximations":[195],"correlation":[197],"bounds":[198],"against":[199],"majority":[201],"function.":[202]},"counts_by_year":[{"year":2022,"cited_by_count":1}],"updated_date":"2026-06-11T09:08:48.828518","created_date":"2025-10-10T00:00:00"}
