{"id":"https://openalex.org/W4416763273","doi":"https://doi.org/10.1145/3776647","title":"Hadamard-Pi: Equational Quantum Programming","display_name":"Hadamard-Pi: Equational Quantum Programming","publication_year":2026,"publication_date":"2026-01-08","ids":{"openalex":"https://openalex.org/W4416763273","doi":"https://doi.org/10.1145/3776647"},"language":"en","primary_location":{"id":"doi:10.1145/3776647","is_oa":true,"landing_page_url":"https://doi.org/10.1145/3776647","pdf_url":null,"source":{"id":"https://openalex.org/S4210216081","display_name":"Proceedings of the ACM on Programming Languages","issn_l":"2475-1421","issn":["2475-1421"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319798","host_organization_name":"Association for Computing Machinery","host_organization_lineage":["https://openalex.org/P4310319798"],"host_organization_lineage_names":["Association for Computing Machinery"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Proceedings of the ACM on Programming Languages","raw_type":"journal-article"},"type":"article","indexed_in":["arxiv","crossref"],"open_access":{"is_oa":true,"oa_status":"diamond","oa_url":"https://doi.org/10.1145/3776647","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5100747363","display_name":"Wang Fang","orcid":"https://orcid.org/0000-0001-7628-1185"},"institutions":[{"id":"https://openalex.org/I98677209","display_name":"University of Edinburgh","ror":"https://ror.org/01nrxwf90","country_code":"GB","type":"education","lineage":["https://openalex.org/I98677209"]}],"countries":["GB"],"is_corresponding":false,"raw_author_name":"Wang Fang","raw_affiliation_strings":["University of Edinburgh, Edinburgh, United Kingdom"],"raw_orcid":"https://orcid.org/0000-0001-7628-1185","affiliations":[{"raw_affiliation_string":"University of Edinburgh, Edinburgh, United Kingdom","institution_ids":["https://openalex.org/I98677209"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5035888204","display_name":"Chris Heunen","orcid":"https://orcid.org/0000-0001-7393-2640"},"institutions":[{"id":"https://openalex.org/I98677209","display_name":"University of Edinburgh","ror":"https://ror.org/01nrxwf90","country_code":"GB","type":"education","lineage":["https://openalex.org/I98677209"]}],"countries":["GB"],"is_corresponding":false,"raw_author_name":"Chris Heunen","raw_affiliation_strings":["University of Edinburgh, Edinburgh, United Kingdom"],"raw_orcid":"https://orcid.org/0000-0001-7393-2640","affiliations":[{"raw_affiliation_string":"University of Edinburgh, Edinburgh, United Kingdom","institution_ids":["https://openalex.org/I98677209"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5048620730","display_name":"Robin Kaarsgaard","orcid":"https://orcid.org/0000-0002-7672-799X"},"institutions":[{"id":"https://openalex.org/I177969490","display_name":"University of Southern Denmark","ror":"https://ror.org/03yrrjy16","country_code":"DK","type":"education","lineage":["https://openalex.org/I177969490"]}],"countries":["DK"],"is_corresponding":false,"raw_author_name":"Robin Kaarsgaard","raw_affiliation_strings":["University of Southern Denmark, Odense, Denmark"],"raw_orcid":"https://orcid.org/0000-0002-7672-799X","affiliations":[{"raw_affiliation_string":"University of Southern Denmark, Odense, Denmark","institution_ids":["https://openalex.org/I177969490"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":2,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":0.0,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":{"value":0.01837846,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"10","issue":"POPL","first_page":"117","last_page":"143"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10682","display_name":"Quantum Computing Algorithms and Architecture","score":0.8894000053405762,"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":0.8894000053405762,"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":0.026100000366568565,"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/T10622","display_name":"Quantum Mechanics and Applications","score":0.009100000374019146,"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/set","display_name":"Set (abstract data type)","score":0.48829999566078186},{"id":"https://openalex.org/keywords/hadamard-transform","display_name":"Hadamard transform","score":0.4814999997615814},{"id":"https://openalex.org/keywords/algebra-over-a-field","display_name":"Algebra over a field","score":0.44209998846054077},{"id":"https://openalex.org/keywords/quantum","display_name":"Quantum","score":0.41350001096725464},{"id":"https://openalex.org/keywords/semantics","display_name":"Semantics (computer science)","score":0.3865000009536743},{"id":"https://openalex.org/keywords/quantum-algorithm","display_name":"Quantum algorithm","score":0.375},{"id":"https://openalex.org/keywords/quantum-computer","display_name":"Quantum computer","score":0.36649999022483826},{"id":"https://openalex.org/keywords/completeness","display_name":"Completeness (order theory)","score":0.36000001430511475}],"concepts":[{"id":"https://openalex.org/C177264268","wikidata":"https://www.wikidata.org/wiki/Q1514741","display_name":"Set (abstract data type)","level":2,"score":0.48829999566078186},{"id":"https://openalex.org/C60292330","wikidata":"https://www.wikidata.org/wiki/Q1014065","display_name":"Hadamard transform","level":2,"score":0.4814999997615814},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.4611000120639801},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.44209998846054077},{"id":"https://openalex.org/C84114770","wikidata":"https://www.wikidata.org/wiki/Q46344","display_name":"Quantum","level":2,"score":0.41350001096725464},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.4043999910354614},{"id":"https://openalex.org/C184337299","wikidata":"https://www.wikidata.org/wiki/Q1437428","display_name":"Semantics (computer science)","level":2,"score":0.3865000009536743},{"id":"https://openalex.org/C137019171","wikidata":"https://www.wikidata.org/wiki/Q2623817","display_name":"Quantum algorithm","level":3,"score":0.375},{"id":"https://openalex.org/C58053490","wikidata":"https://www.wikidata.org/wiki/Q176555","display_name":"Quantum computer","level":3,"score":0.36649999022483826},{"id":"https://openalex.org/C17231256","wikidata":"https://www.wikidata.org/wiki/Q5156540","display_name":"Completeness (order theory)","level":2,"score":0.36000001430511475},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.3456000089645386},{"id":"https://openalex.org/C20274610","wikidata":"https://www.wikidata.org/wiki/Q185410","display_name":"Reversible computing","level":4,"score":0.3400999903678894},{"id":"https://openalex.org/C80444323","wikidata":"https://www.wikidata.org/wiki/Q2878974","display_name":"Theoretical computer science","level":1,"score":0.33629998564720154},{"id":"https://openalex.org/C199360897","wikidata":"https://www.wikidata.org/wiki/Q9143","display_name":"Programming language","level":1,"score":0.3246000111103058},{"id":"https://openalex.org/C188971069","wikidata":"https://www.wikidata.org/wiki/Q1636734","display_name":"Universal algebra","level":3,"score":0.3086000084877014},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.305400013923645},{"id":"https://openalex.org/C128838566","wikidata":"https://www.wikidata.org/wiki/Q275603","display_name":"Logic programming","level":2,"score":0.3043000102043152},{"id":"https://openalex.org/C192061203","wikidata":"https://www.wikidata.org/wiki/Q765620","display_name":"Quantum Turing machine","level":5,"score":0.2953000068664551},{"id":"https://openalex.org/C5274069","wikidata":"https://www.wikidata.org/wiki/Q2285707","display_name":"Categorical variable","level":2,"score":0.28519999980926514},{"id":"https://openalex.org/C102565666","wikidata":"https://www.wikidata.org/wiki/Q1378301","display_name":"Universal set","level":3,"score":0.2734000086784363},{"id":"https://openalex.org/C58849907","wikidata":"https://www.wikidata.org/wiki/Q2118982","display_name":"Quantum gate","level":4,"score":0.27239999175071716},{"id":"https://openalex.org/C20528804","wikidata":"https://www.wikidata.org/wiki/Q997433","display_name":"Dependent type","level":3,"score":0.27140000462532043},{"id":"https://openalex.org/C32241873","wikidata":"https://www.wikidata.org/wiki/Q236975","display_name":"Classical logic","level":2,"score":0.2671999931335449},{"id":"https://openalex.org/C124148022","wikidata":"https://www.wikidata.org/wiki/Q2122210","display_name":"Quantum circuit","level":5,"score":0.2605000138282776},{"id":"https://openalex.org/C154564781","wikidata":"https://www.wikidata.org/wiki/Q5384236","display_name":"Equational logic","level":3,"score":0.25049999356269836}],"mesh":[],"locations_count":5,"locations":[{"id":"doi:10.1145/3776647","is_oa":true,"landing_page_url":"https://doi.org/10.1145/3776647","pdf_url":null,"source":{"id":"https://openalex.org/S4210216081","display_name":"Proceedings of the ACM on Programming Languages","issn_l":"2475-1421","issn":["2475-1421"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319798","host_organization_name":"Association for Computing Machinery","host_organization_lineage":["https://openalex.org/P4310319798"],"host_organization_lineage_names":["Association for Computing Machinery"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Proceedings of the ACM on Programming Languages","raw_type":"journal-article"},{"id":"pmh:oai:sdu.dk:openaire/3cb10832-7fd0-4c43-a7dd-90b161e3d6a5","is_oa":true,"landing_page_url":"https://portal.findresearcher.sdu.dk/da/publications/3cb10832-7fd0-4c43-a7dd-90b161e3d6a5","pdf_url":null,"source":{"id":"https://openalex.org/S4306400423","display_name":"University of Southern Denmark Research Portal (University of Southern Denmark)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I177969490","host_organization_name":"University of Southern Denmark","host_organization_lineage":["https://openalex.org/I177969490"],"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":"Fang, W, Heunen, C & Kaarsgaard, R 2026, 'Hadamard-Pi : Equational Quantum Programming', Proceedings of the ACM on Programming Languages, vol. 10, no. POPL, 5, pp. 117-143. https://doi.org/10.1145/3776647","raw_type":"info:eu-repo/semantics/article"},{"id":"pmh:oai:arXiv.org:2506.06835","is_oa":true,"landing_page_url":"http://arxiv.org/abs/2506.06835","pdf_url":"https://arxiv.org/pdf/2506.06835","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.ed.ac.uk:openaire/9572609d-ad60-43ff-ad60-154a0a372ede","is_oa":true,"landing_page_url":"https://www.research.ed.ac.uk/en/publications/9572609d-ad60-43ff-ad60-154a0a372ede","pdf_url":"https://www.research.ed.ac.uk/files/633455841/FangEtalPACMPL2026HadamardPi.pdf","source":{"id":"https://openalex.org/S4406922455","display_name":"Edinburgh Research Explorer","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","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Fang, W, Heunen, C & Kaarsgaard, R 2026, 'Hadamard-Pi : Equational quantum programming', Proceedings of the ACM on Programming Languages, vol. 10, no. POPL, 5, pp. 117-143. https://doi.org/10.1145/3776647","raw_type":"info:eu-repo/semantics/article"},{"id":"pmh:oai:pure.ed.ac.uk:publications/9572609d-ad60-43ff-ad60-154a0a372ede","is_oa":true,"landing_page_url":"https://hdl.handle.net/20.500.11820/9572609d-ad60-43ff-ad60-154a0a372ede","pdf_url":null,"source":{"id":"https://openalex.org/S4406922455","display_name":"Edinburgh Research Explorer","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","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Fang, W, Heunen, C & Kaarsgaard, R 2026, 'Hadamard-Pi : Equational quantum programming', Proceedings of the ACM on Programming Languages, vol. 10, no. POPL, 5, pp. 117-143. https://doi.org/10.1145/3776647","raw_type":"info:eu-repo/semantics/article"}],"best_oa_location":{"id":"doi:10.1145/3776647","is_oa":true,"landing_page_url":"https://doi.org/10.1145/3776647","pdf_url":null,"source":{"id":"https://openalex.org/S4210216081","display_name":"Proceedings of the ACM on Programming Languages","issn_l":"2475-1421","issn":["2475-1421"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319798","host_organization_name":"Association for Computing Machinery","host_organization_lineage":["https://openalex.org/P4310319798"],"host_organization_lineage_names":["Association for Computing Machinery"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Proceedings of the ACM on Programming Languages","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G3015897474","display_name":"Rubber DUQ: Flexible Dynamic Universal Quantum programming","funder_award_id":"EP/X025551/1","funder_id":"https://openalex.org/F4320334627","funder_display_name":"Engineering and Physical Sciences Research Council"}],"funders":[{"id":"https://openalex.org/F4320334627","display_name":"Engineering and Physical Sciences Research Council","ror":"https://ror.org/0439y7842"}],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":0,"referenced_works":[],"related_works":[],"abstract_inverted_index":{"Quantum":[0],"computing":[1],"offers":[2],"advantages":[3],"over":[4],"classical":[5,39,74],"computation,":[6],"yet":[7],"the":[8,13,19,32,48,72,85,123,131],"precise":[9],"features":[10],"that":[11,99],"set":[12,37],"two":[14],"apart":[15],"remain":[16],"unclear.":[17],"In":[18],"standard":[20],"quantum":[21,45,68],"circuit":[22],"model,":[23],"adding":[24],"a":[25,35,62,66,80,93,103,113,118],"1-qubit":[26],"basis-changing":[27],"gate\u2014commonly":[28],"chosen":[29],"to":[30,84],"be":[31],"Hadamard":[33,86],"gate\u2014to":[34],"universal":[36,44,73],"of":[38,112,125],"reversible":[40,75],"gates":[41],"yields":[42],"computationally":[43],"computation.":[46],"However,":[47],"computational":[49],"behaviours":[50],"enabled":[51],"by":[52,64,102,110],"this":[53],"addition":[54],"are":[55],"not":[56],"fully":[57],"characterised.":[58],"We":[59],"give":[60],"such":[61],"characterisation":[63],"introducing":[65],"small":[67],"programming":[69,76],"language":[70,77,89],"extending":[71],"\u03a0":[78],"with":[79,92,128],"single":[81],"primitive":[82],"corresponding":[83,119],"gate.":[87],"The":[88],"comes":[90],"equipped":[91],"sound":[94],"and":[95,117],"complete":[96],"categorical":[97],"semantics":[98],"is":[100,108],"specified":[101],"purely":[104],"equational":[105],"theory.":[106],"Completeness":[107],"shown":[109],"means":[111],"novel":[114],"finite":[115],"presentation,":[116],"synthesis":[120],"algorithm,":[121],"for":[122],"groups":[124],"orthogonal":[126],"matrices":[127],"entries":[129],"in":[130],"ring":[132],"<mml:math":[133],"xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"":[134],"display=\"inline\">":[135],"<mml:mrow>":[136,142],"<mml:mi>z</mml:mi>":[137],"<mml:mo":[138,148],"stretchy=\"false\">[</mml:mo>":[139],"<mml:mfrac>":[140],"<mml:mn>1</mml:mn>":[141],"<mml:msqrt>":[143],"<mml:mn>2</mml:mn>":[144],"</mml:msqrt>":[145],"</mml:mrow>":[146,150],"</mml:mfrac>":[147],"stretchy=\"false\">]</mml:mo>":[149],"</mml:math>":[151],".":[152]},"counts_by_year":[],"updated_date":"2026-07-26T07:53:14.480251","created_date":"2025-10-10T00:00:00"}
