{"id":"https://openalex.org/W7214165147","doi":"https://doi.org/10.48550/arxiv.2609.28112","title":"Quantum Topological Data Analysis Beyond Betti Numbers: Complexity Hardness $\\&$ An Algorithm for Torsion Witness","display_name":"Quantum Topological Data Analysis Beyond Betti Numbers: Complexity Hardness $\\&$ An Algorithm for Torsion Witness","publication_year":2026,"publication_date":"2026-09-23","ids":{"openalex":"https://openalex.org/W7214165147","doi":"https://doi.org/10.48550/arxiv.2609.28112"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2609.28112","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.28112","pdf_url":"https://arxiv.org/pdf/2609.28112","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/2609.28112","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5034327756","display_name":"Nhat A. Nghiem","orcid":"https://orcid.org/0000-0001-8743-2831","observed_orcids":["https://orcid.org/0000-0001-8743-2831"]},"institutions":[{"id":"https://openalex.org/I181727666","display_name":"University of Economics Ho Chi Minh City","ror":"https://ror.org/010yce376","country_code":"VN","type":"education","lineage":["https://openalex.org/I181727666"]}],"countries":["VN"],"is_corresponding":false,"raw_author_name":"Nghiem, Nhat A.","raw_affiliation_strings":["Institute of Applied Mathematics , University of Economics Ho Chi Minh City"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Institute of Applied Mathematics , University of Economics Ho Chi Minh City","institution_ids":["https://openalex.org/I181727666"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5085151983","display_name":"Dominic William Berry","orcid":"https://orcid.org/0000-0003-3446-1449","observed_orcids":["https://orcid.org/0000-0003-3446-1449"]},"institutions":[{"id":"https://openalex.org/I99043593","display_name":"Macquarie University","ror":"https://ror.org/01sf06y89","country_code":"AU","type":"education","lineage":["https://openalex.org/I99043593"]}],"countries":["AU"],"is_corresponding":false,"raw_author_name":"Berry, Dominic W.","raw_affiliation_strings":["School of Mathematical and Physical Sciences , Macquarie University , Sydney , NSW 2109 , Australia"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Mathematical and Physical Sciences , Macquarie University , Sydney , NSW 2109 , Australia","institution_ids":["https://openalex.org/I99043593"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5101605496","display_name":"Trung V. Phan","orcid":"https://orcid.org/0000-0002-4018-0275","observed_orcids":["https://orcid.org/0000-0002-4018-0275"]},"institutions":[{"id":"https://openalex.org/I1327163397","display_name":"State University of New York","ror":"https://ror.org/01q1z8k08","country_code":"US","type":"education","lineage":["https://openalex.org/I1327163397"]},{"id":"https://openalex.org/I59553526","display_name":"Stony Brook University","ror":"https://ror.org/05qghxh33","country_code":"US","type":"education","lineage":["https://openalex.org/I59553526"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Phan, Trung V.","raw_affiliation_strings":["Department of Natural Sciences , Scripps and Pitzer Colleges ,","Institute for Theoretical Physics , State University of New York at Stony Brook , Stony Brook , NY 11794-3840 , USA"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Natural Sciences , Scripps and Pitzer Colleges ,","institution_ids":[]},{"raw_affiliation_string":"Institute for Theoretical Physics , State University of New York at Stony Brook , Stony Brook , NY 11794-3840 , USA","institution_ids":["https://openalex.org/I1327163397","https://openalex.org/I59553526"]}]}],"institutions":[],"countries_distinct_count":3,"institutions_distinct_count":4,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":true,"cited_by_count":0,"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/T12536","display_name":"Topological and Geometric Data Analysis","score":0.49219998717308044,"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"}},"topics":[{"id":"https://openalex.org/T12536","display_name":"Topological and Geometric Data Analysis","score":0.49219998717308044,"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/T10682","display_name":"Quantum Computing Algorithms and Architecture","score":0.48170000314712524,"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/T10896","display_name":"Homotopy and Cohomology in Algebraic Topology","score":0.006200000178068876,"subfield":{"id":"https://openalex.org/subfields/2610","display_name":"Mathematical Physics"},"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/p-torsion","display_name":"p-torsion","score":0.8100000023841858},{"id":"https://openalex.org/keywords/clique-complexes","display_name":"clique complexes","score":0.7200000286102295},{"id":"https://openalex.org/keywords/np-hardness","display_name":"NP-hardness","score":0.9340000152587891},{"id":"https://openalex.org/keywords/bockstein-homomorphism","display_name":"Bockstein homomorphism","score":0.9190000295639038},{"id":"https://openalex.org/keywords/smith-normal-form","display_name":"Smith normal form","score":0.9869999885559082},{"id":"https://openalex.org/keywords/cohomology","display_name":"cohomology","score":0.7789999842643738},{"id":"https://openalex.org/keywords/quantum-algorithms","display_name":"quantum algorithms","score":0.6349999904632568},{"id":"https://openalex.org/keywords/quantum-advantage","display_name":"quantum advantage","score":0.7540000081062317},{"id":"https://openalex.org/keywords/betti-numbers","display_name":"Betti numbers","score":0.8029999732971191}],"concepts":[{"id":"https://openalex.org/C129621563","wikidata":"https://www.wikidata.org/wiki/Q429593","display_name":"Betti number","level":2,"score":0.6395000219345093},{"id":"https://openalex.org/C77461463","wikidata":"https://www.wikidata.org/wiki/Q7827201","display_name":"Torsion (gastropod)","level":2,"score":0.632099986076355},{"id":"https://openalex.org/C84114770","wikidata":"https://www.wikidata.org/wiki/Q46344","display_name":"Quantum","level":2,"score":0.5777999758720398},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.551800012588501},{"id":"https://openalex.org/C137019171","wikidata":"https://www.wikidata.org/wiki/Q2623817","display_name":"Quantum algorithm","level":3,"score":0.5289000272750854},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.5144000053405762},{"id":"https://openalex.org/C58053490","wikidata":"https://www.wikidata.org/wiki/Q176555","display_name":"Quantum computer","level":3,"score":0.46619999408721924},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.38499999046325684},{"id":"https://openalex.org/C2781204021","wikidata":"https://www.wikidata.org/wiki/Q6497091","display_name":"Lattice (music)","level":2,"score":0.3208000063896179},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.3188000023365021},{"id":"https://openalex.org/C43682372","wikidata":"https://www.wikidata.org/wiki/Q7269080","display_name":"Quantum operation","level":4,"score":0.31380000710487366},{"id":"https://openalex.org/C80444323","wikidata":"https://www.wikidata.org/wiki/Q2878974","display_name":"Theoretical computer science","level":1,"score":0.31290000677108765},{"id":"https://openalex.org/C184720557","wikidata":"https://www.wikidata.org/wiki/Q7825049","display_name":"Topology (electrical circuits)","level":2,"score":0.30390000343322754},{"id":"https://openalex.org/C187929450","wikidata":"https://www.wikidata.org/wiki/Q994399","display_name":"Simplicial complex","level":2,"score":0.30320000648498535},{"id":"https://openalex.org/C62799726","wikidata":"https://www.wikidata.org/wiki/Q190056","display_name":"Hilbert space","level":2,"score":0.2906999886035919},{"id":"https://openalex.org/C84883863","wikidata":"https://www.wikidata.org/wiki/Q7825777","display_name":"Toric code","level":4,"score":0.28839999437332153},{"id":"https://openalex.org/C2778926657","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum system","level":3,"score":0.287200003862381},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.28060001134872437},{"id":"https://openalex.org/C169699857","wikidata":"https://www.wikidata.org/wiki/Q2122243","display_name":"Quantum information","level":3,"score":0.2793000042438507},{"id":"https://openalex.org/C11255438","wikidata":"https://www.wikidata.org/wiki/Q7269085","display_name":"Quantum process","level":4,"score":0.27570000290870667},{"id":"https://openalex.org/C58849907","wikidata":"https://www.wikidata.org/wiki/Q2118982","display_name":"Quantum gate","level":4,"score":0.2750000059604645},{"id":"https://openalex.org/C158693339","wikidata":"https://www.wikidata.org/wiki/Q190524","display_name":"Eigenvalues and eigenvectors","level":2,"score":0.25760000944137573},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.2524999976158142}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2609.28112","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.28112","pdf_url":"https://arxiv.org/pdf/2609.28112","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.2609.28112","doi":"https://doi.org/10.48550/arxiv.2609.28112","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2609.28112","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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2609.28112","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.28112","pdf_url":"https://arxiv.org/pdf/2609.28112","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":[],"funders":[{"id":"https://openalex.org/F4320319003","display_name":"\u0110\u1ea1i h\u1ecdc Kinh t\u1ebf Th\u00e0nh ph\u1ed1 H\u1ed3 Ch\u00ed Minh","ror":"https://ror.org/010yce376"}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7214165147.pdf","grobid_xml":"https://content.openalex.org/works/W7214165147.grobid-xml"},"referenced_works_count":0,"referenced_works":[],"related_works":["https://openalex.org/W7078210670","https://openalex.org/W7215124689","https://openalex.org/W7213222986","https://openalex.org/W4390338740","https://openalex.org/W4414896478","https://openalex.org/W4404321698","https://openalex.org/W3209198297","https://openalex.org/W2949428479","https://openalex.org/W4297411649","https://openalex.org/W4391564220"],"abstract_inverted_index":{"Recent":[0],"advances":[1],"have":[2],"revealed":[3],"an":[4],"interplay":[5],"between":[6],"quantum":[7,15,75,90,173,225,253],"computing":[8],"and":[9,26,58,80,89,96,110,160],"topological":[10],"data":[11],"analysis":[12],"(TDA).":[13],"Most":[14],"TDA":[16],"has":[17,136],"focused":[18],"on":[19,54],"Betti":[20,246,264],"numbers,":[21,247,265],"which":[22,219],"characterize":[23],"the":[24,37,134,152,220,228,233],"connectivity":[25],"``holes''":[27],"of":[28,39,141,167],"a":[29,41,93,121,124,137,142,164,172,178,216,223,249],"dataset.":[30],"Homology,":[31],"however,":[32],"contains":[33,116,198],"additional":[34],"information":[35],"in":[36,65,70,218],"form":[38],"torsion:":[40],"nontrivial":[42],"cycle":[43],"can":[44,266],"become":[45],"trivial":[46],"after":[47],"being":[48],"repeated":[49],"finitely,":[50],"revealing":[51],"global":[52],"constraints":[53],"how":[55],"cycles":[56],"combine":[57],"wrap":[59],"around":[60],"one":[61],"another.":[62],"Beyond":[63],"applications":[64],"biomolecular":[66],"studies,":[67],"torsion":[68,85,180],"appears":[69],"physical":[71],"settings":[72],"including":[73,151],"homological":[74,125],"rotor":[76,126],"codes,":[77],"discrete":[78],"charges,":[79],"gauge":[81],"sectors.":[82],"We":[83,147,214],"study":[84],"from":[86],"both":[87],"classical":[88,230],"perspectives.":[91],"Given":[92],"graph":[94],"$G$":[95],"its":[97,210,263],"clique":[98],"complex":[99],"$K":[100],"=":[101],"\\mathrm{Cl}(G)$,":[102],"we":[103,170],"first":[104],"prove":[105],"that,":[106],"for":[107,163,200,244],"fixed":[108,183],"$r$":[109],"prime":[111],"$p$,":[112],"deciding":[113,132],"whether":[114,133],"$H_r(K,\\mathbb{Z})$":[115,195],"$p$-torsion":[117,199],"is":[118,128,145],"NP-hard.":[119,146],"As":[120],"corollary,":[122],"when":[123],"code":[127,135],"specified":[129],"by":[130],"$G$,":[131],"finite-dimensional":[138],"logical":[139],"sector":[140],"given":[143],"order":[144],"discuss":[148],"related":[149],"problems,":[150],"Bockstein":[153],"homomorphism,":[154],"Smith":[155],"normal":[156],"form,":[157],"lattice":[158],"saturation,":[159],"cohomology.":[161],"Second,":[162],"finite":[165],"set":[166],"primes":[168],"$P$,":[169],"develop":[171],"algorithm":[174,221,231],"that":[175,193,259],"serves":[176],"as":[177],"one-sided":[179],"witness.":[181],"For":[182],"$r$,":[184],"it":[185],"outputs":[186],"WITNESS":[187,191],"or":[188,196,212],"INCONCLUSIVE,":[189],"i.e.":[190],"certifies":[192],"either":[194],"$H_{r-1}(K,\\mathbb{Z})$":[197],"some":[201],"$p\\in":[202],"P$":[203],"while":[204],"INCONCLUSIVE":[205],"makes":[206],"no":[207],"claim":[208],"about":[209],"presence":[211],"absence.":[213],"identify":[215],"regime":[217],"achieves":[222],"near-quadratic":[224],"speedup":[226],"over":[227],"corresponding":[229],"under":[232],"same":[234],"input":[235],"model.":[236],"Our":[237],"NP-hardness":[238],"result":[239],"complements":[240],"recent":[241],"hardness":[242],"results":[243,257],"estimating":[245],"adding":[248],"complexity-theoretic":[250],"perspective":[251],"to":[252],"TDA.":[254],"Together,":[255],"these":[256],"demonstrate":[258],"integral":[260],"homology,":[261],"beyond":[262],"be":[267],"computationally":[268],"challenging.":[269]},"counts_by_year":[],"updated_date":"2026-11-05T12:25:35.769452","created_date":"2026-09-25T00:00:00"}