{"id":"https://openalex.org/W7118755289","doi":"https://doi.org/10.48550/arxiv.2601.02518","title":"Diffusion Computation versus Quantum Computation: A Comparative Model for Order Finding and Factoring","display_name":"Diffusion Computation versus Quantum Computation: A Comparative Model for Order Finding and Factoring","publication_year":2026,"publication_date":"2026-01-05","ids":{"openalex":"https://openalex.org/W7118755289","doi":"https://doi.org/10.48550/arxiv.2601.02518"},"language":null,"primary_location":{"id":"doi:10.48550/arxiv.2601.02518","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2601.02518","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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"article"},"type":"preprint","indexed_in":["datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://doi.org/10.48550/arxiv.2601.02518","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5069420524","display_name":"Carlos A. Cadavid","orcid":"https://orcid.org/0000-0002-7421-7531"},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Cadavid, Carlos A.","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5037588036","display_name":"Paulina Hoyos","orcid":"https://orcid.org/0009-0009-8949-998X"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Hoyos, Paulina","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5014804262","display_name":"Jay Jorgenson","orcid":"https://orcid.org/0000-0002-9765-0944"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Jorgenson, Jay","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5115959982","display_name":"Lejla Smajlovic","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Smajlovi\u0107, Lejla","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5122275815","display_name":"J. D. V\u00e9lez","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"V\u00e9lez, J. D.","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":5,"corresponding_author_ids":["https://openalex.org/A5069420524"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":false,"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/T12056","display_name":"Markov Chains and Monte Carlo Methods","score":0.1834000051021576,"subfield":{"id":"https://openalex.org/subfields/2613","display_name":"Statistics and Probability"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T12056","display_name":"Markov Chains and Monte Carlo Methods","score":0.1834000051021576,"subfield":{"id":"https://openalex.org/subfields/2613","display_name":"Statistics and Probability"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11612","display_name":"Stochastic Gradient Optimization Techniques","score":0.17069999873638153,"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/T10682","display_name":"Quantum Computing Algorithms and Architecture","score":0.16529999673366547,"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/factorization","display_name":"Factorization","score":0.6428999900817871},{"id":"https://openalex.org/keywords/multiplicative-function","display_name":"Multiplicative function","score":0.5351999998092651},{"id":"https://openalex.org/keywords/prime-factor","display_name":"Prime factor","score":0.4912000000476837},{"id":"https://openalex.org/keywords/prime","display_name":"Prime (order theory)","score":0.4366999864578247},{"id":"https://openalex.org/keywords/parameterized-complexity","display_name":"Parameterized complexity","score":0.43160000443458557},{"id":"https://openalex.org/keywords/unitary-state","display_name":"Unitary state","score":0.37869998812675476},{"id":"https://openalex.org/keywords/bipartite-graph","display_name":"Bipartite graph","score":0.3716000020503998},{"id":"https://openalex.org/keywords/unitary-matrix","display_name":"Unitary matrix","score":0.3707999885082245},{"id":"https://openalex.org/keywords/vertex","display_name":"Vertex (graph theory)","score":0.36959999799728394},{"id":"https://openalex.org/keywords/integer","display_name":"Integer (computer science)","score":0.3671000003814697}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.724399983882904},{"id":"https://openalex.org/C187834632","wikidata":"https://www.wikidata.org/wiki/Q188804","display_name":"Factorization","level":2,"score":0.6428999900817871},{"id":"https://openalex.org/C42747912","wikidata":"https://www.wikidata.org/wiki/Q1048447","display_name":"Multiplicative function","level":2,"score":0.5351999998092651},{"id":"https://openalex.org/C30860621","wikidata":"https://www.wikidata.org/wiki/Q49008","display_name":"Prime factor","level":3,"score":0.4912000000476837},{"id":"https://openalex.org/C184992742","wikidata":"https://www.wikidata.org/wiki/Q7243229","display_name":"Prime (order theory)","level":2,"score":0.4366999864578247},{"id":"https://openalex.org/C165464430","wikidata":"https://www.wikidata.org/wiki/Q1570441","display_name":"Parameterized complexity","level":2,"score":0.43160000443458557},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.4097000062465668},{"id":"https://openalex.org/C67820243","wikidata":"https://www.wikidata.org/wiki/Q179164","display_name":"Unitary state","level":2,"score":0.37869998812675476},{"id":"https://openalex.org/C197657726","wikidata":"https://www.wikidata.org/wiki/Q174733","display_name":"Bipartite graph","level":3,"score":0.3716000020503998},{"id":"https://openalex.org/C96214315","wikidata":"https://www.wikidata.org/wiki/Q727103","display_name":"Unitary matrix","level":3,"score":0.3707999885082245},{"id":"https://openalex.org/C80899671","wikidata":"https://www.wikidata.org/wiki/Q1304193","display_name":"Vertex (graph theory)","level":3,"score":0.36959999799728394},{"id":"https://openalex.org/C97137487","wikidata":"https://www.wikidata.org/wiki/Q729138","display_name":"Integer (computer science)","level":2,"score":0.3671000003814697},{"id":"https://openalex.org/C45374587","wikidata":"https://www.wikidata.org/wiki/Q12525525","display_name":"Computation","level":2,"score":0.35929998755455017},{"id":"https://openalex.org/C17349429","wikidata":"https://www.wikidata.org/wiki/Q1049914","display_name":"Matrix multiplication","level":3,"score":0.34610000252723694},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.3425999879837036},{"id":"https://openalex.org/C58053490","wikidata":"https://www.wikidata.org/wiki/Q176555","display_name":"Quantum computer","level":3,"score":0.34049999713897705},{"id":"https://openalex.org/C106487976","wikidata":"https://www.wikidata.org/wiki/Q685816","display_name":"Matrix (chemical analysis)","level":2,"score":0.33649998903274536},{"id":"https://openalex.org/C42355184","wikidata":"https://www.wikidata.org/wiki/Q1361088","display_name":"Matrix decomposition","level":3,"score":0.33059999346733093},{"id":"https://openalex.org/C182306322","wikidata":"https://www.wikidata.org/wiki/Q1779371","display_name":"Order (exchange)","level":2,"score":0.3294000029563904},{"id":"https://openalex.org/C137559481","wikidata":"https://www.wikidata.org/wiki/Q1231787","display_name":"Dixon's factorization method","level":5,"score":0.3255999982357025},{"id":"https://openalex.org/C2781311116","wikidata":"https://www.wikidata.org/wiki/Q83306","display_name":"Group (periodic table)","level":2,"score":0.31459999084472656},{"id":"https://openalex.org/C2778023678","wikidata":"https://www.wikidata.org/wiki/Q554403","display_name":"Duality (order theory)","level":2,"score":0.30410000681877136},{"id":"https://openalex.org/C177225278","wikidata":"https://www.wikidata.org/wiki/Q192674","display_name":"Factoring","level":2,"score":0.2913999855518341},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.2913999855518341},{"id":"https://openalex.org/C132525143","wikidata":"https://www.wikidata.org/wiki/Q141488","display_name":"Graph","level":2,"score":0.28119999170303345},{"id":"https://openalex.org/C177264268","wikidata":"https://www.wikidata.org/wiki/Q1514741","display_name":"Set (abstract data type)","level":2,"score":0.2703999876976013},{"id":"https://openalex.org/C2777735758","wikidata":"https://www.wikidata.org/wiki/Q817765","display_name":"Path (computing)","level":2,"score":0.26820001006126404},{"id":"https://openalex.org/C98763669","wikidata":"https://www.wikidata.org/wiki/Q176645","display_name":"Markov chain","level":2,"score":0.2662000060081482},{"id":"https://openalex.org/C311688","wikidata":"https://www.wikidata.org/wiki/Q2393193","display_name":"Time complexity","level":2,"score":0.257099986076355},{"id":"https://openalex.org/C111335779","wikidata":"https://www.wikidata.org/wiki/Q3454686","display_name":"Reduction (mathematics)","level":2,"score":0.25369998812675476},{"id":"https://openalex.org/C162392398","wikidata":"https://www.wikidata.org/wiki/Q272404","display_name":"Finite set","level":2,"score":0.25220000743865967},{"id":"https://openalex.org/C48103436","wikidata":"https://www.wikidata.org/wiki/Q599031","display_name":"State (computer science)","level":2,"score":0.25189998745918274},{"id":"https://openalex.org/C13355873","wikidata":"https://www.wikidata.org/wiki/Q2920850","display_name":"Connection (principal bundle)","level":2,"score":0.2515999972820282}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.48550/arxiv.2601.02518","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2601.02518","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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":null,"raw_source_name":null,"raw_type":"article"}],"best_oa_location":{"id":"doi:10.48550/arxiv.2601.02518","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2601.02518","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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"article"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":0,"referenced_works":[],"related_works":[],"abstract_inverted_index":{"We":[0,82,150,201],"study":[1],"a":[2,23,27,37,70,89,135,173],"hybrid":[3],"computational":[4],"model":[5,155],"for":[6,111],"integer":[7,64],"factorization":[8,175],"in":[9,87,214],"which":[10,65],"the":[11,97,107,157,166,183],"only":[12,181],"non-classical":[13],"resource":[14],"is":[15,30,66,96,197],"access":[16],"to":[17,32,45,106,162,171],"an":[18,46,52,62,117,123,147],"\\emph{iterated":[19],"diffusion":[20,144,221],"process}":[21],"on":[22,182],"finite":[24],"graph.":[25],"Concretely,":[26],"\\emph{diffusion":[28],"step}":[29],"defined":[31],"be":[33,61,132],"one":[34],"application":[35],"of":[36,55,168,186,190,230],"symmetric":[38],"stochastic":[39],"matrix":[40],"(the":[41],"half-lazy":[42],"walk":[43],"operator)":[44],"$\\ell^{1}$--normalized":[47],"state":[48],"vector,":[49],"followed":[50],"by":[51,206],"optional":[53],"readout":[54],"selected":[56],"coordinates.":[57],"Let":[58],"$N\\ge":[59],"3$":[60],"odd":[63,77],"neither":[67],"prime":[68,71,188],"nor":[69],"power,":[72],"and":[73,102,199,210,220,228],"let":[74],"$b\\in(\\mathbb{Z}/N\\mathbb{Z})^\\ast$":[75],"have":[76],"multiplicative":[78],"order":[79,163],"$r={\\rm":[80],"ord}_N(b)$.":[81],"construct,":[83],"without":[84],"knowing":[85],"$r$":[86,130],"advance,":[88],"weighted":[90],"Cayley":[91],"graph":[92],"whose":[93,103,177],"vertex":[94],"set":[95],"cyclic":[98],"subgroup":[99],"$\\langle":[100],"b\\rangle$":[101],"edges":[104],"correspond":[105],"powers":[108],"$b^{\\pm":[109],"2^t}$":[110],"$t\\le":[112],"\\lfloor":[113],"\\log_2":[114],"N\\rfloor+1$.":[115],"Using":[116],"explicit":[118],"spectral":[119],"decomposition":[120],"together":[121],"with":[122,146,156,194],"elementary":[124],"doubling":[125],"lemma,":[126],"we":[127,211,224],"show":[128],"that":[129],"can":[131],"recovered":[133],"from":[134,160],"single":[136],"heat-kernel":[137],"value":[138],"after":[139],"at":[140],"most":[141],"$O((\\log_2":[142],"N)^2)$":[143],"steps,":[145],"effective":[148],"bound.":[149],"then":[151],"combine":[152],"this":[153],"order-finding":[154],"standard":[158],"reduction":[159],"factoring":[161],"finding":[164],"(in":[165],"spirit":[167],"Shor's":[169,195],"framework)":[170],"obtain":[172],"randomized":[174],"procedure":[176],"success":[178],"probability":[179],"depends":[180],"number":[184],"$m$":[185],"distinct":[187],"factors":[189],"$N$.":[191],"Our":[192],"comparison":[193],"algorithm":[196],"\\emph{conceptual":[198],"model-based}.":[200],"replace":[202],"unitary":[203],"$\\ell^2$":[204],"evolution":[205],"Markovian":[207],"$\\ell^1$":[208],"evolution,":[209],"report":[212],"complexity":[213],"two":[215],"cost":[216],"measures:":[217],"digital":[218],"steps":[219],"steps.":[222],"Finally,":[223],"include":[225],"illustrative":[226],"examples":[227],"discussion":[229],"practical":[231],"implementations.":[232]},"counts_by_year":[],"updated_date":"2026-05-03T08:25:01.440150","created_date":"2026-01-08T00:00:00"}