{"id":"https://openalex.org/W4415489893","doi":"https://doi.org/10.3390/a18110678","title":"A Hybrid Quantum\u2013Classical Spectral Solver for Nonlinear Differential Equations","display_name":"A Hybrid Quantum\u2013Classical Spectral Solver for Nonlinear Differential Equations","publication_year":2025,"publication_date":"2025-10-23","ids":{"openalex":"https://openalex.org/W4415489893","doi":"https://doi.org/10.3390/a18110678"},"language":"en","primary_location":{"id":"doi:10.3390/a18110678","is_oa":true,"landing_page_url":"https://doi.org/10.3390/a18110678","pdf_url":"https://www.mdpi.com/1999-4893/18/11/678/pdf?version=1761228677","source":{"id":"https://openalex.org/S190629608","display_name":"Algorithms","issn_l":"1999-4893","issn":["1999-4893"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Algorithms","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://www.mdpi.com/1999-4893/18/11/678/pdf?version=1761228677","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5087447078","display_name":"Samar Aseeri","orcid":null},"institutions":[{"id":"https://openalex.org/I71920554","display_name":"King Abdullah University of Science and Technology","ror":"https://ror.org/01q3tbs38","country_code":"SA","type":"education","lineage":["https://openalex.org/I71920554"]}],"countries":["SA"],"is_corresponding":true,"raw_author_name":"Samar A. Aseeri","raw_affiliation_strings":["King Abdullah University of Science and Technology, Thuwal 23955, Saudi Arabia"],"affiliations":[{"raw_affiliation_string":"King Abdullah University of Science and Technology, Thuwal 23955, Saudi Arabia","institution_ids":["https://openalex.org/I71920554"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5087447078"],"corresponding_institution_ids":["https://openalex.org/I71920554"],"apc_list":{"value":1400,"currency":"CHF","value_usd":1515},"apc_paid":{"value":1400,"currency":"CHF","value_usd":1515},"fwci":0.0,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":{"value":0.16574821,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"18","issue":"11","first_page":"678","last_page":"678"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10020","display_name":"Quantum Information and Cryptography","score":0.9939000010490417,"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/T10020","display_name":"Quantum Information and Cryptography","score":0.9939000010490417,"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":0.9932000041007996,"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/T11416","display_name":"Numerical methods for differential equations","score":0.9922000169754028,"subfield":{"id":"https://openalex.org/subfields/2612","display_name":"Numerical Analysis"},"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/solver","display_name":"Solver","score":0.6938999891281128},{"id":"https://openalex.org/keywords/quantum","display_name":"Quantum","score":0.5942999720573425},{"id":"https://openalex.org/keywords/quantum-computer","display_name":"Quantum computer","score":0.5296000242233276},{"id":"https://openalex.org/keywords/nonlinear-system","display_name":"Nonlinear system","score":0.5102999806404114},{"id":"https://openalex.org/keywords/residual","display_name":"Residual","score":0.4724999964237213},{"id":"https://openalex.org/keywords/binary-number","display_name":"Binary number","score":0.47200000286102295},{"id":"https://openalex.org/keywords/speedup","display_name":"Speedup","score":0.4472000002861023},{"id":"https://openalex.org/keywords/singular-value-decomposition","display_name":"Singular value decomposition","score":0.40700000524520874},{"id":"https://openalex.org/keywords/quantum-circuit","display_name":"Quantum circuit","score":0.3720000088214874},{"id":"https://openalex.org/keywords/chebyshev-polynomials","display_name":"Chebyshev polynomials","score":0.3714999854564667}],"concepts":[{"id":"https://openalex.org/C2778770139","wikidata":"https://www.wikidata.org/wiki/Q1966904","display_name":"Solver","level":2,"score":0.6938999891281128},{"id":"https://openalex.org/C84114770","wikidata":"https://www.wikidata.org/wiki/Q46344","display_name":"Quantum","level":2,"score":0.5942999720573425},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.5867000222206116},{"id":"https://openalex.org/C58053490","wikidata":"https://www.wikidata.org/wiki/Q176555","display_name":"Quantum computer","level":3,"score":0.5296000242233276},{"id":"https://openalex.org/C158622935","wikidata":"https://www.wikidata.org/wiki/Q660848","display_name":"Nonlinear system","level":2,"score":0.5102999806404114},{"id":"https://openalex.org/C155512373","wikidata":"https://www.wikidata.org/wiki/Q287450","display_name":"Residual","level":2,"score":0.4724999964237213},{"id":"https://openalex.org/C48372109","wikidata":"https://www.wikidata.org/wiki/Q3913","display_name":"Binary number","level":2,"score":0.47200000286102295},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.4708000123500824},{"id":"https://openalex.org/C68339613","wikidata":"https://www.wikidata.org/wiki/Q1549489","display_name":"Speedup","level":2,"score":0.4472000002861023},{"id":"https://openalex.org/C22789450","wikidata":"https://www.wikidata.org/wiki/Q420904","display_name":"Singular value decomposition","level":2,"score":0.40700000524520874},{"id":"https://openalex.org/C124148022","wikidata":"https://www.wikidata.org/wiki/Q2122210","display_name":"Quantum circuit","level":5,"score":0.3720000088214874},{"id":"https://openalex.org/C129785596","wikidata":"https://www.wikidata.org/wiki/Q619511","display_name":"Chebyshev polynomials","level":2,"score":0.3714999854564667},{"id":"https://openalex.org/C137019171","wikidata":"https://www.wikidata.org/wiki/Q2623817","display_name":"Quantum algorithm","level":3,"score":0.36230000853538513},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.35659998655319214},{"id":"https://openalex.org/C182310444","wikidata":"https://www.wikidata.org/wiki/Q1332643","display_name":"Boundary value problem","level":2,"score":0.3449999988079071},{"id":"https://openalex.org/C203087015","wikidata":"https://www.wikidata.org/wiki/Q378201","display_name":"Qubit","level":3,"score":0.34130001068115234},{"id":"https://openalex.org/C155332342","wikidata":"https://www.wikidata.org/wiki/Q1432976","display_name":"Generalized minimal residual method","level":3,"score":0.33660000562667847},{"id":"https://openalex.org/C177179195","wikidata":"https://www.wikidata.org/wiki/Q7268372","display_name":"Quadratic unconstrained binary optimization","level":4,"score":0.3292999863624573},{"id":"https://openalex.org/C129844170","wikidata":"https://www.wikidata.org/wiki/Q41299","display_name":"Quadratic equation","level":2,"score":0.32919999957084656},{"id":"https://openalex.org/C78045399","wikidata":"https://www.wikidata.org/wiki/Q11214","display_name":"Differential equation","level":2,"score":0.32589998841285706},{"id":"https://openalex.org/C23463724","wikidata":"https://www.wikidata.org/wiki/Q2308831","display_name":"Spectral method","level":2,"score":0.3231000006198883},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.3174999952316284},{"id":"https://openalex.org/C34862557","wikidata":"https://www.wikidata.org/wiki/Q178985","display_name":"Ode","level":2,"score":0.31290000677108765},{"id":"https://openalex.org/C134146338","wikidata":"https://www.wikidata.org/wiki/Q1815901","display_name":"Electronic circuit","level":2,"score":0.3125999867916107},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.3043999969959259},{"id":"https://openalex.org/C51544822","wikidata":"https://www.wikidata.org/wiki/Q465274","display_name":"Ordinary differential equation","level":3,"score":0.29409998655319214},{"id":"https://openalex.org/C159694833","wikidata":"https://www.wikidata.org/wiki/Q2321565","display_name":"Iterative method","level":2,"score":0.28380000591278076},{"id":"https://openalex.org/C3770464","wikidata":"https://www.wikidata.org/wiki/Q775963","display_name":"Smoothing","level":2,"score":0.28369998931884766},{"id":"https://openalex.org/C93779851","wikidata":"https://www.wikidata.org/wiki/Q271977","display_name":"Partial differential equation","level":2,"score":0.28279998898506165},{"id":"https://openalex.org/C185429906","wikidata":"https://www.wikidata.org/wiki/Q1130160","display_name":"Estimator","level":2,"score":0.2808000147342682},{"id":"https://openalex.org/C2781204021","wikidata":"https://www.wikidata.org/wiki/Q6497091","display_name":"Lattice (music)","level":2,"score":0.27619999647140503},{"id":"https://openalex.org/C2778926657","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum system","level":3,"score":0.27390000224113464},{"id":"https://openalex.org/C147764199","wikidata":"https://www.wikidata.org/wiki/Q6865248","display_name":"Minification","level":2,"score":0.2732999920845032},{"id":"https://openalex.org/C15706264","wikidata":"https://www.wikidata.org/wiki/Q230883","display_name":"Quantum state","level":3,"score":0.2703999876976013},{"id":"https://openalex.org/C187455244","wikidata":"https://www.wikidata.org/wiki/Q942353","display_name":"Boolean function","level":2,"score":0.26759999990463257},{"id":"https://openalex.org/C198880260","wikidata":"https://www.wikidata.org/wiki/Q5289813","display_name":"Domain decomposition methods","level":3,"score":0.2667999863624573}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.3390/a18110678","is_oa":true,"landing_page_url":"https://doi.org/10.3390/a18110678","pdf_url":"https://www.mdpi.com/1999-4893/18/11/678/pdf?version=1761228677","source":{"id":"https://openalex.org/S190629608","display_name":"Algorithms","issn_l":"1999-4893","issn":["1999-4893"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Algorithms","raw_type":"journal-article"},{"id":"pmh:oai:doaj.org/article:da3c5d633d5d4a2caef37dcdf65c920c","is_oa":true,"landing_page_url":"https://doaj.org/article/da3c5d633d5d4a2caef37dcdf65c920c","pdf_url":null,"source":{"id":"https://openalex.org/S112646816","display_name":"SHILAP Revista de lepidopterolog\u00eda","issn_l":"0300-5267","issn":["0300-5267","2340-4078"],"is_oa":true,"is_in_doaj":true,"is_core":false,"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":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Algorithms, Vol 18, Iss 11, p 678 (2025)","raw_type":"article"}],"best_oa_location":{"id":"doi:10.3390/a18110678","is_oa":true,"landing_page_url":"https://doi.org/10.3390/a18110678","pdf_url":"https://www.mdpi.com/1999-4893/18/11/678/pdf?version=1761228677","source":{"id":"https://openalex.org/S190629608","display_name":"Algorithms","issn_l":"1999-4893","issn":["1999-4893"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Algorithms","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W4415489893.pdf","grobid_xml":"https://content.openalex.org/works/W4415489893.grobid-xml"},"referenced_works_count":21,"referenced_works":["https://openalex.org/W1490180844","https://openalex.org/W1975286851","https://openalex.org/W2584608327","https://openalex.org/W2781738013","https://openalex.org/W2963198496","https://openalex.org/W3034967390","https://openalex.org/W3108818881","https://openalex.org/W3119836548","https://openalex.org/W3123005213","https://openalex.org/W3132684784","https://openalex.org/W4283528683","https://openalex.org/W4386785058","https://openalex.org/W4389560601","https://openalex.org/W4399625233","https://openalex.org/W4399887488","https://openalex.org/W4402473907","https://openalex.org/W4403325745","https://openalex.org/W4403333745","https://openalex.org/W4403909320","https://openalex.org/W4409483184","https://openalex.org/W4409495403"],"related_works":[],"abstract_inverted_index":{"We":[0,103],"investigate":[1],"hybrid":[2,175],"quantum\u2013classical":[3],"solvers":[4],"for":[5,92,183,199,210],"nonlinear":[6],"boundary":[7,50],"value":[8],"problems":[9],"using":[10,62],"Chebyshev":[11],"spectral":[12,28,128],"collocation.":[13],"Unlike":[14],"prior":[15],"methods":[16],"such":[17],"as":[18],"H\u2013DES,":[19],"which":[20,74,89],"repeatedly":[21],"recompile":[22],"circuits":[23,64],"and":[24,80,111,161,171,178],"encode":[25],"the":[26,31,38,78,120,124,133,137,166,184],"entire":[27],"basis":[29],"on":[30,60,94,109,115],"quantum":[32,43,96,197,212],"processor,":[33],"our":[34],"framework":[35],"offloads":[36],"only":[37],"residual":[39,58],"minimisation":[40,59],"to":[41,65,157],"a":[42,67,82,95,100,106,146,152,180,189,203],"backend":[44],"while":[45],"retaining":[46],"classical":[47,121,192],"enforcement":[48],"of":[49,169],"conditions.":[51],"Two":[52],"paradigms":[53],"are":[54],"considered:":[55],"(i)":[56],"gate-based":[57,141,172],"CUDA-Q":[61,110],"variational":[63,148],"evaluate":[66],"Cubic":[68],"Unconstrained":[69,84],"Binary":[70,85],"Optimisation":[71,86],"(CUBO)":[72],"cost,":[73],"naturally":[75],"arises":[76],"from":[77],"discretisation,":[79],"(ii)":[81],"Quadratic":[83],"(QUBO)":[87],"reformulation,":[88],"is":[90],"required":[91],"execution":[93],"annealer,":[97],"executed":[98],"via":[99],"classical\u2013quantum":[101],"mapping.":[102],"further":[104],"explore":[105],"CUBO":[107,142],"extension":[108],"direct":[112],"residual-to-energy":[113],"mapping":[114],"annealers.":[116],"Benchmarks":[117],"confirm":[118],"that":[119],"solver":[122,143],"reproduces":[123],"analytic":[125],"solution":[126],"with":[127,207],"accuracy;":[129],"among":[130],"quantum-enhanced":[131],"methods,":[132],"annealer-based":[134],"QUBO":[135],"yields":[136],"closest":[138],"approximation.":[139],"The":[140],"improves":[144],"upon":[145],"legacy":[147],"baseline":[149],"but":[150],"exhibits":[151],"small":[153],"interior":[154],"bias":[155],"due":[156],"limited":[158],"circuit":[159],"depth":[160],"precision.":[162],"These":[163],"findings":[164],"underscore":[165],"complementary":[167],"roles":[168],"annealers":[170],"devices":[173],"in":[174,196],"scientific":[176],"computing":[177],"demonstrate":[179],"feasible":[181],"workflow":[182],"NISQ":[185],"era":[186],"rather":[187],"than":[188],"speedup":[190],"over":[191],"methods.":[193],"Recent":[194],"progress":[195],"algorithms":[198],"differential":[200],"equations":[201],"signals":[202],"rapidly":[204],"maturing":[205],"field":[206],"significant":[208],"potential":[209],"practical":[211],"advantage.":[213]},"counts_by_year":[],"updated_date":"2026-03-13T14:20:09.374765","created_date":"2025-10-24T00:00:00"}