{"id":"https://openalex.org/W7212048159","doi":"https://doi.org/10.48550/arxiv.2609.07579","title":"Geometry Dependence of Error Thresholds in Two-Dimensional Toric Codes","display_name":"Geometry Dependence of Error Thresholds in Two-Dimensional Toric Codes","publication_year":2026,"publication_date":"2026-09-07","ids":{"openalex":"https://openalex.org/W7212048159","doi":"https://doi.org/10.48550/arxiv.2609.07579"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2609.07579","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.07579","pdf_url":"https://arxiv.org/pdf/2609.07579","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,"listed_in":[],"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"},"volume":null,"type":"preprint","is_traditional":true,"indexed_in":["arxiv","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/2609.07579","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5080287843","display_name":"D. Lessing","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I181369854","display_name":"Friedrich-Alexander-Universit\u00e4t Erlangen-N\u00fcrnberg","ror":"https://ror.org/00f7hpc57","country_code":"DE","type":"education","lineage":["https://openalex.org/I181369854"]}],"countries":["DE"],"is_corresponding":false,"raw_author_name":"Lessing, D.","raw_affiliation_strings":["Lehrstuhl f\u00fcr Theoretische Physik V , Friedrich-Alexander-Universit\u00e4t Erlangen-N\u00fcrnberg , Staudtstr. 7 , 91058 Erlangen , Germany"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Lehrstuhl f\u00fcr Theoretische Physik V , Friedrich-Alexander-Universit\u00e4t Erlangen-N\u00fcrnberg , Staudtstr. 7 , 91058 Erlangen , Germany","institution_ids":["https://openalex.org/I181369854"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5004495790","display_name":"Anja Langheld","orcid":"https://orcid.org/0000-0003-2638-8827","observed_orcids":["https://orcid.org/0000-0003-2638-8827"]},"institutions":[{"id":"https://openalex.org/I181369854","display_name":"Friedrich-Alexander-Universit\u00e4t Erlangen-N\u00fcrnberg","ror":"https://ror.org/00f7hpc57","country_code":"DE","type":"education","lineage":["https://openalex.org/I181369854"]}],"countries":["DE"],"is_corresponding":false,"raw_author_name":"Langheld, A.","raw_affiliation_strings":["Lehrstuhl f\u00fcr Theoretische Physik V , Friedrich-Alexander-Universit\u00e4t Erlangen-N\u00fcrnberg , Staudtstr. 7 , 91058 Erlangen , Germany"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Lehrstuhl f\u00fcr Theoretische Physik V , Friedrich-Alexander-Universit\u00e4t Erlangen-N\u00fcrnberg , Staudtstr. 7 , 91058 Erlangen , Germany","institution_ids":["https://openalex.org/I181369854"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5151024446","display_name":"C. Kr\u00e4mer","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I181369854","display_name":"Friedrich-Alexander-Universit\u00e4t Erlangen-N\u00fcrnberg","ror":"https://ror.org/00f7hpc57","country_code":"DE","type":"education","lineage":["https://openalex.org/I181369854"]}],"countries":["DE"],"is_corresponding":false,"raw_author_name":"Kr\u00e4mer, C.","raw_affiliation_strings":["Lehrstuhl f\u00fcr Theoretische Physik V , Friedrich-Alexander-Universit\u00e4t Erlangen-N\u00fcrnberg , Staudtstr. 7 , 91058 Erlangen , Germany"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Lehrstuhl f\u00fcr Theoretische Physik V , Friedrich-Alexander-Universit\u00e4t Erlangen-N\u00fcrnberg , Staudtstr. 7 , 91058 Erlangen , Germany","institution_ids":["https://openalex.org/I181369854"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5027065532","display_name":"Jan Alexander Koziol","orcid":"https://orcid.org/0000-0003-2464-7709","observed_orcids":["https://orcid.org/0000-0003-2464-7709"]},"institutions":[{"id":"https://openalex.org/I129774422","display_name":"University of Vienna","ror":"https://ror.org/03prydq77","country_code":"AT","type":"education","lineage":["https://openalex.org/I129774422"]}],"countries":["AT"],"is_corresponding":false,"raw_author_name":"Koziol, J. A.","raw_affiliation_strings":["Faculty of Physics , University of Vienna , Boltzmanngasse 5 , 1090 Vienna , Austria"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Faculty of Physics , University of Vienna , Boltzmanngasse 5 , 1090 Vienna , Austria","institution_ids":["https://openalex.org/I129774422"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5150982123","display_name":"K. P. Schmidt","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I181369854","display_name":"Friedrich-Alexander-Universit\u00e4t Erlangen-N\u00fcrnberg","ror":"https://ror.org/00f7hpc57","country_code":"DE","type":"education","lineage":["https://openalex.org/I181369854"]}],"countries":["DE"],"is_corresponding":false,"raw_author_name":"Schmidt, K. P.","raw_affiliation_strings":["Lehrstuhl f\u00fcr Theoretische Physik V , Friedrich-Alexander-Universit\u00e4t Erlangen-N\u00fcrnberg , Staudtstr. 7 , 91058 Erlangen , Germany"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Lehrstuhl f\u00fcr Theoretische Physik V , Friedrich-Alexander-Universit\u00e4t Erlangen-N\u00fcrnberg , Staudtstr. 7 , 91058 Erlangen , Germany","institution_ids":["https://openalex.org/I181369854"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":2,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":true,"cited_by_count":1,"citation_normalized_percentile":null,"cited_by_percentile_year":null,"biblio":{"volume":null,"issue":null,"first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10682","display_name":"Quantum Computing Algorithms and Architecture","score":0.9678000211715698,"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":0.9678000211715698,"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/T11804","display_name":"Quantum many-body systems","score":0.008700000122189522,"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"}},{"id":"https://openalex.org/T10591","display_name":"Theoretical and Computational Physics","score":0.00800000037997961,"subfield":{"id":"https://openalex.org/subfields/3104","display_name":"Condensed Matter Physics"},"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/toric-code","display_name":"toric code","score":0.8859999775886536},{"id":"https://openalex.org/keywords/error-threshold","display_name":"error threshold","score":0.7689999938011169},{"id":"https://openalex.org/keywords/lattice-geometry","display_name":"lattice geometry","score":0.7860000133514404},{"id":"https://openalex.org/keywords/random-bond-ising-model","display_name":"random bond Ising model","score":0.9950000047683716},{"id":"https://openalex.org/keywords/nishimori-line","display_name":"Nishimori line","score":0.9290000200271606},{"id":"https://openalex.org/keywords/maximum-likelihood-decoding","display_name":"maximum likelihood decoding","score":0.7149999737739563},{"id":"https://openalex.org/keywords/bit-flip-errors","display_name":"bit-flip errors","score":0.7360000014305115},{"id":"https://openalex.org/keywords/phase-flip-errors","display_name":"phase-flip errors","score":1.0},{"id":"https://openalex.org/keywords/lattice-duality","display_name":"lattice duality","score":0.8349999785423279},{"id":"https://openalex.org/keywords/kagome-lattice","display_name":"kagome lattice","score":0.7609999775886536},{"id":"https://openalex.org/keywords/honeycomb-lattice","display_name":"honeycomb lattice","score":0.8740000128746033},{"id":"https://openalex.org/keywords/dice-lattice","display_name":"dice lattice","score":0.8389999866485596},{"id":"https://openalex.org/keywords/triangular-lattice","display_name":"triangular lattice","score":0.7829999923706055},{"id":"https://openalex.org/keywords/square-lattice","display_name":"square lattice","score":0.8090000152587891},{"id":"https://openalex.org/keywords/coordination-number","display_name":"coordination number","score":0.6769999861717224},{"id":"https://openalex.org/keywords/quantum-error-correction","display_name":"quantum error correction","score":0.6430000066757202},{"id":"https://openalex.org/keywords/code-capacity","display_name":"code capacity","score":0.8019999861717224}],"concepts":[{"id":"https://openalex.org/C2781204021","wikidata":"https://www.wikidata.org/wiki/Q6497091","display_name":"Lattice (music)","level":2,"score":0.7020000219345093},{"id":"https://openalex.org/C51329190","wikidata":"https://www.wikidata.org/wiki/Q1076349","display_name":"Ising model","level":2,"score":0.6291000247001648},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.5372999906539917},{"id":"https://openalex.org/C84883863","wikidata":"https://www.wikidata.org/wiki/Q7825777","display_name":"Toric code","level":4,"score":0.4788999855518341},{"id":"https://openalex.org/C19499675","wikidata":"https://www.wikidata.org/wiki/Q232207","display_name":"Monte Carlo method","level":2,"score":0.43810001015663147},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.3553999960422516},{"id":"https://openalex.org/C45374587","wikidata":"https://www.wikidata.org/wiki/Q12525525","display_name":"Computation","level":2,"score":0.3490999937057495},{"id":"https://openalex.org/C189645446","wikidata":"https://www.wikidata.org/wiki/Q350865","display_name":"Mirroring","level":2,"score":0.3452000021934509},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.34310001134872437},{"id":"https://openalex.org/C121864883","wikidata":"https://www.wikidata.org/wiki/Q677916","display_name":"Statistical physics","level":1,"score":0.33660000562667847},{"id":"https://openalex.org/C103088060","wikidata":"https://www.wikidata.org/wiki/Q1062839","display_name":"Error detection and correction","level":2,"score":0.3345000147819519},{"id":"https://openalex.org/C2776760102","wikidata":"https://www.wikidata.org/wiki/Q5139990","display_name":"Code (set theory)","level":3,"score":0.29840001463890076},{"id":"https://openalex.org/C57273362","wikidata":"https://www.wikidata.org/wiki/Q576722","display_name":"Decoding methods","level":2,"score":0.2964000105857849},{"id":"https://openalex.org/C2777894999","wikidata":"https://www.wikidata.org/wiki/Q4781758","display_name":"Approx","level":2,"score":0.26969999074935913},{"id":"https://openalex.org/C147764199","wikidata":"https://www.wikidata.org/wiki/Q6865248","display_name":"Minification","level":2,"score":0.262800008058548},{"id":"https://openalex.org/C158843486","wikidata":"https://www.wikidata.org/wiki/Q2137810","display_name":"Complex geometry","level":2,"score":0.2587999999523163},{"id":"https://openalex.org/C198352243","wikidata":"https://www.wikidata.org/wiki/Q37105","display_name":"Line (geometry)","level":2,"score":0.25200000405311584}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2609.07579","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.07579","pdf_url":"https://arxiv.org/pdf/2609.07579","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,"listed_in":[],"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":"doi:10.48550/arxiv.2609.07579","doi":"https://doi.org/10.48550/arxiv.2609.07579","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2609.07579","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,"listed_in":[],"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":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2609.07579","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.07579","pdf_url":"https://arxiv.org/pdf/2609.07579","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,"listed_in":[],"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"},"sustainable_development_goals":[],"study_designs":[],"awards":[{"id":"https://openalex.org/G5302217128","display_name":null,"funder_award_id":"10.55776/COE 1","funder_id":"https://openalex.org/F4320321181","funder_display_name":"Austrian Science Fund"},{"id":"https://openalex.org/G8431032053","display_name":"Neutral-atom quantum systems with high connectivity","funder_award_id":"F101200","funder_id":"https://openalex.org/F4320321181","funder_display_name":"Austrian Science Fund"},{"id":"https://openalex.org/G8590477180","display_name":null,"funder_award_id":"10.55776","funder_id":"https://openalex.org/F4320321181","funder_display_name":"Austrian Science Fund"}],"funders":[{"id":"https://openalex.org/F4320320300","display_name":"European Commission","ror":"https://ror.org/00k4n6c32"},{"id":"https://openalex.org/F4320320873","display_name":"Friedrich-Alexander-Universit\u00e4t Erlangen-N\u00fcrnberg","ror":"https://ror.org/00f7hpc57"},{"id":"https://openalex.org/F4320320879","display_name":"Deutsche Forschungsgemeinschaft","ror":"https://ror.org/018mejw64"},{"id":"https://openalex.org/F4320321181","display_name":"Austrian Science Fund","ror":"https://ror.org/013tf3c58"}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7212048159.pdf","grobid_xml":"https://content.openalex.org/works/W7212048159.grobid-xml"},"referenced_works_count":44,"referenced_works":["https://openalex.org/W1479964525","https://openalex.org/W1527520754","https://openalex.org/W1542370521","https://openalex.org/W1964277505","https://openalex.org/W1984021697","https://openalex.org/W2002007522","https://openalex.org/W2002664623","https://openalex.org/W2007762738","https://openalex.org/W2013754283","https://openalex.org/W2017686710","https://openalex.org/W2018671186","https://openalex.org/W2021251923","https://openalex.org/W2032367804","https://openalex.org/W2040443882","https://openalex.org/W2040834288","https://openalex.org/W2049471987","https://openalex.org/W2064795418","https://openalex.org/W2070210144","https://openalex.org/W2075431083","https://openalex.org/W2085706156","https://openalex.org/W2088636071","https://openalex.org/W2107453273","https://openalex.org/W2114207424","https://openalex.org/W2128164182","https://openalex.org/W2131690084","https://openalex.org/W2135273380","https://openalex.org/W2155808716","https://openalex.org/W2166065655","https://openalex.org/W2170171286","https://openalex.org/W2179731956","https://openalex.org/W2755710612","https://openalex.org/W2982169647","https://openalex.org/W3038010021","https://openalex.org/W3148422733","https://openalex.org/W4226134739","https://openalex.org/W4321610982","https://openalex.org/W4387542025","https://openalex.org/W4389373471","https://openalex.org/W4392182417","https://openalex.org/W4405183524","https://openalex.org/W4415150927","https://openalex.org/W4415272443","https://openalex.org/W4416075932","https://openalex.org/W7163864962"],"related_works":["https://openalex.org/W7171372187","https://openalex.org/W1991443278","https://openalex.org/W7214145599","https://openalex.org/W7171498534","https://openalex.org/W7152038002","https://openalex.org/W2513693704","https://openalex.org/W4392270597","https://openalex.org/W4232410326","https://openalex.org/W2206517764","https://openalex.org/W2991596793"],"abstract_inverted_index":{"We":[0,33,78,117],"investigate":[1],"how":[2],"the":[3,7,15,22,27,35,39,64,72,85,89,101,111,120,132,136,144,149,153,160],"lattice":[4,54,99,107],"geometry":[5],"influences":[6],"error":[8,82,154],"thresholds":[9,51,83,122],"of":[10,17,38,67,123,163],"two-dimensional":[11,28],"toric":[12,23],"codes.":[13],"In":[14],"presence":[16],"bit-":[18],"or":[19],"phase-flip":[20,102],"errors,":[21],"code":[24,75],"maps":[25],"onto":[26],"random-bond":[29],"Ising":[30],"model":[31],"(RBIM).":[32],"determine":[34],"critical":[36],"behaviour":[37,138],"RBIM":[40],"using":[41],"replica-exchange":[42],"Monte":[43],"Carlo":[44],"simulations.":[45],"While":[46,143],"previous":[47],"studies":[48],"have":[49],"explored":[50],"for":[52,88,140],"general":[53],"geometries":[55],"under":[56],"suboptimal":[57,141],"decoders":[58,69],"such":[59],"as":[60],"minimum-weight":[61],"perfect":[62],"matching,":[63],"rapid":[65],"development":[66],"near-optimal":[68],"makes":[70],"resolving":[71],"ultimate,":[73],"maximum-likelihood":[74],"capacity":[76],"relevant.":[77],"compute":[79],"these":[80],"optimal":[81,121],"on":[84,104,114,152],"Nishimori":[86],"line":[87],"square,":[90],"honeycomb,":[91],"triangular,":[92],"dice,":[93],"and":[94,165],"kagome":[95],"lattices.":[96],"Owing":[97],"to":[98,110],"duality,":[100],"threshold":[103,113],"a":[105,127,169],"given":[106],"is":[108],"equivalent":[109],"bit-flip":[112],"its":[115,174],"dual.":[116],"find":[118],"that":[119,159],"dual-lattice":[124],"pairs":[125],"display":[126],"characteristic":[128],"duality-driven":[129],"splitting":[130],"around":[131],"self-dual":[133],"square-lattice,":[134],"mirroring":[135],"qualitative":[137],"observed":[139],"decoders.":[142],"average":[145],"coordination":[146],"number":[147],"has":[148],"strongest":[150],"impact":[151],"threshold,":[155],"our":[156],"results":[157],"demonstrate":[158],"detailed":[161],"arrangement":[162],"vertices":[164],"plaquettes":[166],"also":[167],"plays":[168],"significant":[170],"role":[171],"in":[172],"determining":[173],"precise":[175],"value.":[176]},"counts_by_year":[{"year":2026,"cited_by_count":1}],"updated_date":"2026-11-07T11:25:07.637180","created_date":"2026-09-10T00:00:00"}