{"id":"https://openalex.org/W3135453135","doi":"https://doi.org/10.32408/compositionality-5-5","title":"Completeness of the ZH-calculus","display_name":"Completeness of the ZH-calculus","publication_year":2023,"publication_date":"2023-07-12","ids":{"openalex":"https://openalex.org/W3135453135","doi":"https://doi.org/10.32408/compositionality-5-5","mag":"3135453135"},"language":"en","primary_location":{"id":"doi:10.32408/compositionality-5-5","is_oa":true,"landing_page_url":"https://doi.org/10.32408/compositionality-5-5","pdf_url":"https://compositionality-journal.org/papers/compositionality-5-5/pdf/","source":{"id":"https://openalex.org/S4210182901","display_name":"Compositionality","issn_l":"2631-4444","issn":["2631-4444"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Compositionality","raw_type":"journal-article"},"type":"article","indexed_in":["arxiv","crossref"],"open_access":{"is_oa":true,"oa_status":"diamond","oa_url":"https://compositionality-journal.org/papers/compositionality-5-5/pdf/","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5029698011","display_name":"Miriam Backens","orcid":"https://orcid.org/0000-0002-5418-1084"},"institutions":[{"id":"https://openalex.org/I79619799","display_name":"University of Birmingham","ror":"https://ror.org/03angcq70","country_code":"GB","type":"education","lineage":["https://openalex.org/I79619799"]}],"countries":["GB"],"is_corresponding":true,"raw_author_name":"Miriam Backens","raw_affiliation_strings":["School of Computer Science, University of Birmingham, Edgbaston, Birmingham B15 2TT, UK"],"affiliations":[{"raw_affiliation_string":"School of Computer Science, University of Birmingham, Edgbaston, Birmingham B15 2TT, UK","institution_ids":["https://openalex.org/I79619799"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5023841122","display_name":"Aleks Kissinger","orcid":"https://orcid.org/0000-0002-6090-9684"},"institutions":[{"id":"https://openalex.org/I40120149","display_name":"University of Oxford","ror":"https://ror.org/052gg0110","country_code":"GB","type":"education","lineage":["https://openalex.org/I40120149"]}],"countries":["GB"],"is_corresponding":false,"raw_author_name":"Aleks Kissinger","raw_affiliation_strings":["Department of Computer Science, University of Oxford, Wolfson Building, Parks Road, Oxford OX1 3QD, UK"],"affiliations":[{"raw_affiliation_string":"Department of Computer Science, University of Oxford, Wolfson Building, Parks Road, Oxford OX1 3QD, UK","institution_ids":["https://openalex.org/I40120149"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5006538416","display_name":"Hector Miller-Bakewell","orcid":"https://orcid.org/0000-0002-2741-6714"},"institutions":[{"id":"https://openalex.org/I40120149","display_name":"University of Oxford","ror":"https://ror.org/052gg0110","country_code":"GB","type":"education","lineage":["https://openalex.org/I40120149"]}],"countries":["GB"],"is_corresponding":false,"raw_author_name":"Hector Miller-Bakewell","raw_affiliation_strings":["Department of Computer Science, University of Oxford, Wolfson Building, Parks Road, Oxford OX1 3QD, UK"],"affiliations":[{"raw_affiliation_string":"Department of Computer Science, University of Oxford, Wolfson Building, Parks Road, Oxford OX1 3QD, UK","institution_ids":["https://openalex.org/I40120149"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5040545926","display_name":"John van de Wetering","orcid":"https://orcid.org/0000-0002-5405-8959"},"institutions":[{"id":"https://openalex.org/I40120149","display_name":"University of Oxford","ror":"https://ror.org/052gg0110","country_code":"GB","type":"education","lineage":["https://openalex.org/I40120149"]}],"countries":["GB"],"is_corresponding":false,"raw_author_name":"John van de Wetering","raw_affiliation_strings":["Department of Computer Science, University of Oxford, Wolfson Building, Parks Road, Oxford OX1 3QD, UK","Institute for Computing and Information Sciences, Radboud Universiteit, Toernooiveld 212, 6525 EC Nijmegen, NL"],"affiliations":[{"raw_affiliation_string":"Department of Computer Science, University of Oxford, Wolfson Building, Parks Road, Oxford OX1 3QD, UK","institution_ids":["https://openalex.org/I40120149"]},{"raw_affiliation_string":"Institute for Computing and Information Sciences, Radboud Universiteit, Toernooiveld 212, 6525 EC Nijmegen, NL","institution_ids":[]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5057375451","display_name":"Sal Wolffs","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Sal Wolffs","raw_affiliation_strings":["Institute for Computing and Information Sciences, Radboud Universiteit, Toernooiveld 212, 6525 EC Nijmegen, NL"],"affiliations":[{"raw_affiliation_string":"Institute for Computing and Information Sciences, Radboud Universiteit, Toernooiveld 212, 6525 EC Nijmegen, NL","institution_ids":[]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":5,"corresponding_author_ids":["https://openalex.org/A5029698011"],"corresponding_institution_ids":["https://openalex.org/I79619799"],"apc_list":{"value":0,"currency":"USD","value_usd":0},"apc_paid":null,"fwci":0.7032,"has_fulltext":true,"cited_by_count":7,"citation_normalized_percentile":{"value":0.74151068,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":96},"biblio":{"volume":"5","issue":null,"first_page":"5","last_page":"5"},"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/T10020","display_name":"Quantum Information and Cryptography","score":0.9932000041007996,"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/T10126","display_name":"Logic, programming, and type systems","score":0.9925000071525574,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/toffoli-gate","display_name":"Toffoli gate","score":0.9317916035652161},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.6358532905578613},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.4968567192554474},{"id":"https://openalex.org/keywords/qubit","display_name":"Qubit","score":0.33213502168655396},{"id":"https://openalex.org/keywords/calculus","display_name":"Calculus (dental)","score":0.3314630389213562},{"id":"https://openalex.org/keywords/quantum-gate","display_name":"Quantum gate","score":0.23064365983009338},{"id":"https://openalex.org/keywords/quantum","display_name":"Quantum","score":0.15430942177772522},{"id":"https://openalex.org/keywords/quantum-mechanics","display_name":"Quantum mechanics","score":0.12732920050621033},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.09086516499519348}],"concepts":[{"id":"https://openalex.org/C142465778","wikidata":"https://www.wikidata.org/wiki/Q2502619","display_name":"Toffoli gate","level":5,"score":0.9317916035652161},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.6358532905578613},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.4968567192554474},{"id":"https://openalex.org/C203087015","wikidata":"https://www.wikidata.org/wiki/Q378201","display_name":"Qubit","level":3,"score":0.33213502168655396},{"id":"https://openalex.org/C2777686260","wikidata":"https://www.wikidata.org/wiki/Q144037","display_name":"Calculus (dental)","level":2,"score":0.3314630389213562},{"id":"https://openalex.org/C58849907","wikidata":"https://www.wikidata.org/wiki/Q2118982","display_name":"Quantum gate","level":4,"score":0.23064365983009338},{"id":"https://openalex.org/C84114770","wikidata":"https://www.wikidata.org/wiki/Q46344","display_name":"Quantum","level":2,"score":0.15430942177772522},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.12732920050621033},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.09086516499519348},{"id":"https://openalex.org/C199343813","wikidata":"https://www.wikidata.org/wiki/Q12128","display_name":"Dentistry","level":1,"score":0.0},{"id":"https://openalex.org/C71924100","wikidata":"https://www.wikidata.org/wiki/Q11190","display_name":"Medicine","level":0,"score":0.0}],"mesh":[],"locations_count":6,"locations":[{"id":"doi:10.32408/compositionality-5-5","is_oa":true,"landing_page_url":"https://doi.org/10.32408/compositionality-5-5","pdf_url":"https://compositionality-journal.org/papers/compositionality-5-5/pdf/","source":{"id":"https://openalex.org/S4210182901","display_name":"Compositionality","issn_l":"2631-4444","issn":["2631-4444"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Compositionality","raw_type":"journal-article"},{"id":"pmh:oai:dare.uva.nl:publications/3d347f08-55d2-40f5-903e-920c31aca983","is_oa":true,"landing_page_url":"https://handle.uba.uva.nl/personal/pure/en/publications/completeness-of-the-zhcalculus(3d347f08-55d2-40f5-903e-920c31aca983).html","pdf_url":null,"source":{"id":"https://openalex.org/S4306400088","display_name":"UvA-DARE (University of Amsterdam)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I887064364","host_organization_name":"University of Amsterdam","host_organization_lineage":["https://openalex.org/I887064364"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Backens, M, Kissinger, A, Miller-Bakewell, H, van de Wetering, J & Wolffs, S 2023, 'Completeness of the ZH-calculus', Compositionality, vol. 5, 13524. https://doi.org/10.32408/compositionality-5-5","raw_type":"info:eu-repo/semantics/publishedVersion"},{"id":"pmh:oai:arXiv.org:2103.06610","is_oa":true,"landing_page_url":"http://arxiv.org/abs/2103.06610","pdf_url":"https://arxiv.org/pdf/2103.06610","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":"pmh:oai:pure.atira.dk:openaire_cris_publications/006c60e4-6e6b-4a98-bf48-065da398f192","is_oa":true,"landing_page_url":"https://research.birmingham.ac.uk/en/publications/006c60e4-6e6b-4a98-bf48-065da398f192","pdf_url":"https://pure-oai.bham.ac.uk/ws/files/200045182/BackensM2023Completeness.pdf","source":{"id":"https://openalex.org/S4306402634","display_name":"University of Birmingham Research Portal (University of Birmingham)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I79619799","host_organization_name":"University of Birmingham","host_organization_lineage":["https://openalex.org/I79619799"],"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":"Backens, M, Kissinger, A, Miller-Bakewell, H, van de Wetering, J & Wolffs, S 2023, 'Completeness of the ZH-calculus', Compositionality, vol. 5, no. 5. https://doi.org/10.32408/compositionality-5-5","raw_type":"article"},{"id":"pmh:oai:repository.ubn.ru.nl:2066/298006","is_oa":true,"landing_page_url":"https://repository.ubn.ru.nl/handle/2066/298006","pdf_url":"https://repository.ubn.ru.nl//bitstream/handle/2066/298006/298006.pdf","source":{"id":"https://openalex.org/S4306401067","display_name":"Radboud Repository (Radboud University)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I145872427","host_organization_name":"Radboud University Nijmegen","host_organization_lineage":["https://openalex.org/I145872427"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Compositionality, 5, 5, pp. 1-68","raw_type":"Article / Letter to editor"},{"id":"pmh:oai:dare.uva.nl:openaire/3d347f08-55d2-40f5-903e-920c31aca983","is_oa":true,"landing_page_url":"https://hdl.handle.net/11245.1/3d347f08-55d2-40f5-903e-920c31aca983","pdf_url":null,"source":{"id":"https://openalex.org/S4306400088","display_name":"UvA-DARE (University of Amsterdam)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I887064364","host_organization_name":"University of Amsterdam","host_organization_lineage":["https://openalex.org/I887064364"],"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":"Backens, M, Kissinger, A, Miller-Bakewell, H, van de Wetering, J & Wolffs, S 2023, 'Completeness of the ZH-calculus', Compositionality, vol. 5, 13524. https://doi.org/10.32408/compositionality-5-5","raw_type":"info:eu-repo/semantics/publishedVersion"}],"best_oa_location":{"id":"doi:10.32408/compositionality-5-5","is_oa":true,"landing_page_url":"https://doi.org/10.32408/compositionality-5-5","pdf_url":"https://compositionality-journal.org/papers/compositionality-5-5/pdf/","source":{"id":"https://openalex.org/S4210182901","display_name":"Compositionality","issn_l":"2631-4444","issn":["2631-4444"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Compositionality","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G1764870811","display_name":null,"funder_award_id":"FA2386-18-1-4028","funder_id":"https://openalex.org/F4320338279","funder_display_name":"Air Force Office of Scientific Research"},{"id":"https://openalex.org/G2689612763","display_name":null,"funder_award_id":"Marie","funder_id":"https://openalex.org/F4320320300","funder_display_name":"European Commission"},{"id":"https://openalex.org/G3591598716","display_name":null,"funder_award_id":"FA2386","funder_id":"https://openalex.org/F4320338279","funder_display_name":"Air Force Office of Scientific Research"},{"id":"https://openalex.org/G4956428346","display_name":null,"funder_award_id":"Horizon 2020 research and innovatio","funder_id":"https://openalex.org/F4320320300","funder_display_name":"European Commission"},{"id":"https://openalex.org/G5036817778","display_name":null,"funder_award_id":"European Union's Horizon 2020 research and innov","funder_id":"https://openalex.org/F4320320300","funder_display_name":"European Commission"},{"id":"https://openalex.org/G629491556","display_name":null,"funder_award_id":"(NWO)","funder_id":"https://openalex.org/F4320321800","funder_display_name":"Nederlandse Organisatie voor Wetenschappelijk Onderzoek"},{"id":"https://openalex.org/G7589102738","display_name":"Diagrammatic Quantum Computation","funder_award_id":"101018390","funder_id":"https://openalex.org/F4320320300","funder_display_name":"European Commission"},{"id":"https://openalex.org/G8316287456","display_name":null,"funder_award_id":"Rubicon","funder_id":"https://openalex.org/F4320321800","funder_display_name":"Nederlandse Organisatie voor Wetenschappelijk Onderzoek"},{"id":"https://openalex.org/G8318064016","display_name":null,"funder_award_id":"Horizon","funder_id":"https://openalex.org/F4320320300","funder_display_name":"European Commission"},{"id":"https://openalex.org/G8633428685","display_name":null,"funder_award_id":"European Union's Horizon 2020 research and innovat","funder_id":"https://openalex.org/F4320320300","funder_display_name":"European Commission"}],"funders":[{"id":"https://openalex.org/F4320320300","display_name":"European Commission","ror":"https://ror.org/00k4n6c32"},{"id":"https://openalex.org/F4320321800","display_name":"Nederlandse Organisatie voor Wetenschappelijk Onderzoek","ror":"https://ror.org/04jsz6e67"},{"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/W3135453135.pdf"},"referenced_works_count":73,"referenced_works":["https://openalex.org/W1486765721","https://openalex.org/W1617295181","https://openalex.org/W1631356911","https://openalex.org/W1675244148","https://openalex.org/W1925988739","https://openalex.org/W1982928587","https://openalex.org/W2005170092","https://openalex.org/W2015179126","https://openalex.org/W2030867075","https://openalex.org/W2052146120","https://openalex.org/W2081566119","https://openalex.org/W2096331363","https://openalex.org/W2102162525","https://openalex.org/W2119180894","https://openalex.org/W2559582198","https://openalex.org/W2598985404","https://openalex.org/W2610243179","https://openalex.org/W2726846376","https://openalex.org/W2758270599","https://openalex.org/W2784637752","https://openalex.org/W2792699004","https://openalex.org/W2796162739","https://openalex.org/W2798910219","https://openalex.org/W2802560682","https://openalex.org/W2803889673","https://openalex.org/W2901329658","https://openalex.org/W2918135161","https://openalex.org/W2922906818","https://openalex.org/W2938297835","https://openalex.org/W2939312767","https://openalex.org/W2949295853","https://openalex.org/W2950223894","https://openalex.org/W2952708034","https://openalex.org/W2964268018","https://openalex.org/W2966200829","https://openalex.org/W2966537942","https://openalex.org/W2967973346","https://openalex.org/W2987687226","https://openalex.org/W2990961515","https://openalex.org/W3009801593","https://openalex.org/W3012943077","https://openalex.org/W3013980468","https://openalex.org/W3015280770","https://openalex.org/W3045951809","https://openalex.org/W3048149130","https://openalex.org/W3091879340","https://openalex.org/W3099344125","https://openalex.org/W3099455211","https://openalex.org/W3100779651","https://openalex.org/W3100952059","https://openalex.org/W3103164351","https://openalex.org/W3103485545","https://openalex.org/W3103730115","https://openalex.org/W3104500086","https://openalex.org/W3104863176","https://openalex.org/W3105603580","https://openalex.org/W3105893726","https://openalex.org/W3106198031","https://openalex.org/W3109224720","https://openalex.org/W3121796938","https://openalex.org/W3127751074","https://openalex.org/W3141495992","https://openalex.org/W3174817955","https://openalex.org/W3194215427","https://openalex.org/W3196458710","https://openalex.org/W3198777318","https://openalex.org/W4286859498","https://openalex.org/W4287548337","https://openalex.org/W4287644590","https://openalex.org/W4294770912","https://openalex.org/W4297824220","https://openalex.org/W4382132565","https://openalex.org/W6966803337"],"related_works":["https://openalex.org/W2748952813","https://openalex.org/W2772794421","https://openalex.org/W2529811130","https://openalex.org/W3094850919","https://openalex.org/W2888416745","https://openalex.org/W4378942754","https://openalex.org/W2065951251","https://openalex.org/W2282918090","https://openalex.org/W4221155753","https://openalex.org/W4377865206"],"abstract_inverted_index":{"There":[0],"are":[1],"various":[2],"gate":[3,26,73],"sets":[4],"used":[5],"for":[6,38,66,121],"describing":[7,44],"quantum":[8,60],"computation.":[9],"A":[10],"particularly":[11],"popular":[12],"one":[13],"consists":[14],"of":[15,41,57,59,95,103,117,154],"Clifford":[16],"gates":[17,105],"and":[18,106,124],"arbitrary":[19],"single-qubit":[20],"phase":[21],"gates.":[22],"Computations":[23],"in":[24,54],"this":[25,85,122],"set":[27,74,116],"can":[28],"be":[29],"elegantly":[30],"described":[31],"by":[32],"the":[33,81,89,143,155],"ZX-calculus,":[34],"a":[35,39,55,115,174],"graphical":[36,93],"language":[37,94],"class":[40],"string":[42,96],"diagrams":[43,97],"linear":[45],"maps":[46],"between":[47],"qubits.":[48],"The":[49],"ZX-calculus":[50],"has":[51],"proven":[52],"useful":[53],"variety":[56],"areas":[58],"information,":[61],"but":[62],"is":[63,127,158,172],"less":[64],"suitable":[65],"reasoning":[67],"about":[68],"operations":[69,79],"outside":[70],"its":[71],"natural":[72],"such":[75],"as":[76],"multi-linear":[77],"Boolean":[78,110],"like":[80],"Toffoli":[82,104],"gate.":[83],"In":[84],"paper":[86],"we":[87,149],"study":[88],"ZH-calculus,":[90],"an":[91,151],"alternative":[92],"that":[98,157],"does":[99],"allow":[100],"straightforward":[101],"encoding":[102],"other":[107],"more":[108],"complicated":[109],"logic":[111],"circuits.":[112],"We":[113],"find":[114],"simple":[118],"rewrite":[119],"rules":[120],"calculus":[123],"show":[125],"it":[126],"complete":[128,159],"with":[129,160],"respect":[130,161],"to":[131,142,162],"matrices":[132,163],"over":[133,164],"<mml:math":[134,167,170],"xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mrow":[135],"class=\"MJX-TeXAtom-ORD\"><mml:mi":[136],"mathvariant=\"double-struck\">Z</mml:mi></mml:mrow><mml:mo":[137],"stretchy=\"false\">[</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mn>2</mml:mn></mml:mfrac><mml:mo":[138],"stretchy=\"false\">]</mml:mo></mml:math>,":[139],"which":[140],"correspond":[141],"approximately":[144],"universal":[145],"Toffoli+Hadamard":[146],"gateset.":[147],"Furthermore,":[148],"construct":[150],"extended":[152],"version":[153],"ZH-calculus":[156],"any":[165],"ring":[166],"xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>R</mml:mi></mml:math>":[168],"where":[169],"xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:math>":[171],"not":[173],"zero-divisor.":[175]},"counts_by_year":[{"year":2025,"cited_by_count":2},{"year":2024,"cited_by_count":1},{"year":2023,"cited_by_count":1},{"year":2022,"cited_by_count":2},{"year":2020,"cited_by_count":1}],"updated_date":"2026-03-20T23:20:44.827607","created_date":"2025-10-10T00:00:00"}