{"id":"https://openalex.org/W7214722356","doi":"https://doi.org/10.48550/arxiv.2609.33777","title":"Scaling Theory for Learning Low-Energy Quantum Subspaces","display_name":"Scaling Theory for Learning Low-Energy Quantum Subspaces","publication_year":2026,"publication_date":"2026-09-27","ids":{"openalex":"https://openalex.org/W7214722356","doi":"https://doi.org/10.48550/arxiv.2609.33777"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2609.33777","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.33777","pdf_url":"https://arxiv.org/pdf/2609.33777","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/2609.33777","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5102806085","display_name":"Guijing Duan","orcid":"https://orcid.org/0009-0009-2913-7581","observed_orcids":["https://orcid.org/0009-0009-2913-7581"]},"institutions":[{"id":"https://openalex.org/I99065089","display_name":"Tsinghua University","ror":"https://ror.org/03cve4549","country_code":"CN","type":"education","lineage":["https://openalex.org/I99065089"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Duan, Guijing","raw_affiliation_strings":["Department of Physics , Tsinghua University , Beijing 100084 , China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Physics , Tsinghua University , Beijing 100084 , China","institution_ids":["https://openalex.org/I99065089"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5132769166","display_name":"Chen Mo","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I99065089","display_name":"Tsinghua University","ror":"https://ror.org/03cve4549","country_code":"CN","type":"education","lineage":["https://openalex.org/I99065089"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Mo, Chen","raw_affiliation_strings":["Department of Physics , Tsinghua University , Beijing 100084 , China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Physics , Tsinghua University , Beijing 100084 , China","institution_ids":["https://openalex.org/I99065089"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5031223524","display_name":"Di Luo","orcid":"https://orcid.org/0000-0001-6562-1762","observed_orcids":["https://orcid.org/0000-0001-6562-1762"]},"institutions":[{"id":"https://openalex.org/I99065089","display_name":"Tsinghua University","ror":"https://ror.org/03cve4549","country_code":"CN","type":"education","lineage":["https://openalex.org/I99065089"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Luo, Di","raw_affiliation_strings":["Department of Physics , Tsinghua University , Beijing 100084 , China","Institute of Advanced Study , Tsinghua University , Beijing 100084 , China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Physics , Tsinghua University , Beijing 100084 , China","institution_ids":["https://openalex.org/I99065089"]},{"raw_affiliation_string":"Institute of Advanced Study , Tsinghua University , Beijing 100084 , China","institution_ids":["https://openalex.org/I99065089"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":[],"corresponding_institution_ids":["https://openalex.org/I99065089"],"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/T11804","display_name":"Quantum many-body systems","score":0.9965999722480774,"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"}},"topics":[{"id":"https://openalex.org/T11804","display_name":"Quantum many-body systems","score":0.9965999722480774,"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/T10682","display_name":"Quantum Computing Algorithms and Architecture","score":0.0017000000225380063,"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/T10591","display_name":"Theoretical and Computational Physics","score":0.0005000000237487257,"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/quantum-phase-diagram","display_name":"quantum phase diagram","score":0.46299999952316284},{"id":"https://openalex.org/keywords/energy-surface","display_name":"energy surface","score":0.5630000233650208},{"id":"https://openalex.org/keywords/locality","display_name":"locality","score":0.3659999966621399},{"id":"https://openalex.org/keywords/heisenberg-spin-chain","display_name":"Heisenberg spin chain","score":0.6840000152587891},{"id":"https://openalex.org/keywords/transverse-field-ising-chain","display_name":"transverse-field Ising chain","score":0.9879999756813049},{"id":"https://openalex.org/keywords/quantum-many-body-systems","display_name":"quantum many-body systems","score":0.5400000214576721}],"concepts":[{"id":"https://openalex.org/C99844830","wikidata":"https://www.wikidata.org/wiki/Q102441924","display_name":"Scaling","level":2,"score":0.7419000267982483},{"id":"https://openalex.org/C51329190","wikidata":"https://www.wikidata.org/wiki/Q1076349","display_name":"Ising model","level":2,"score":0.5895000100135803},{"id":"https://openalex.org/C34388435","wikidata":"https://www.wikidata.org/wiki/Q2267362","display_name":"Bounded function","level":2,"score":0.5856999754905701},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.5687000155448914},{"id":"https://openalex.org/C12362212","wikidata":"https://www.wikidata.org/wiki/Q728435","display_name":"Linear subspace","level":2,"score":0.557200014591217},{"id":"https://openalex.org/C32834561","wikidata":"https://www.wikidata.org/wiki/Q660730","display_name":"Subspace topology","level":2,"score":0.5238000154495239},{"id":"https://openalex.org/C121864883","wikidata":"https://www.wikidata.org/wiki/Q677916","display_name":"Statistical physics","level":1,"score":0.49129998683929443},{"id":"https://openalex.org/C33676613","wikidata":"https://www.wikidata.org/wiki/Q13415176","display_name":"Dimension (graph theory)","level":2,"score":0.48820000886917114},{"id":"https://openalex.org/C140779682","wikidata":"https://www.wikidata.org/wiki/Q210868","display_name":"Sampling (signal processing)","level":3,"score":0.4702000021934509},{"id":"https://openalex.org/C158693339","wikidata":"https://www.wikidata.org/wiki/Q190524","display_name":"Eigenvalues and eigenvectors","level":2,"score":0.4092000126838684},{"id":"https://openalex.org/C84114770","wikidata":"https://www.wikidata.org/wiki/Q46344","display_name":"Quantum","level":2,"score":0.3937000036239624},{"id":"https://openalex.org/C156778621","wikidata":"https://www.wikidata.org/wiki/Q1365748","display_name":"Spectrum (functional analysis)","level":2,"score":0.3686999976634979},{"id":"https://openalex.org/C28719098","wikidata":"https://www.wikidata.org/wiki/Q44946","display_name":"Point (geometry)","level":2,"score":0.361299991607666},{"id":"https://openalex.org/C177264268","wikidata":"https://www.wikidata.org/wiki/Q1514741","display_name":"Set (abstract data type)","level":2,"score":0.3467000126838684},{"id":"https://openalex.org/C48103436","wikidata":"https://www.wikidata.org/wiki/Q599031","display_name":"State (computer science)","level":2,"score":0.3391999900341034},{"id":"https://openalex.org/C186370098","wikidata":"https://www.wikidata.org/wiki/Q442787","display_name":"Energy (signal processing)","level":2,"score":0.32339999079704285},{"id":"https://openalex.org/C149288129","wikidata":"https://www.wikidata.org/wiki/Q185357","display_name":"Phase transition","level":2,"score":0.32179999351501465},{"id":"https://openalex.org/C69523127","wikidata":"https://www.wikidata.org/wiki/Q4480008","display_name":"Ground state","level":2,"score":0.31610000133514404},{"id":"https://openalex.org/C44280652","wikidata":"https://www.wikidata.org/wiki/Q104837","display_name":"Phase (matter)","level":2,"score":0.3066999912261963},{"id":"https://openalex.org/C32848918","wikidata":"https://www.wikidata.org/wiki/Q845789","display_name":"Observable","level":2,"score":0.30469998717308044},{"id":"https://openalex.org/C61445026","wikidata":"https://www.wikidata.org/wiki/Q217608","display_name":"Fixed point","level":2,"score":0.29600000381469727},{"id":"https://openalex.org/C30732413","wikidata":"https://www.wikidata.org/wiki/Q17092636","display_name":"Intrinsic dimension","level":3,"score":0.2930000126361847},{"id":"https://openalex.org/C151342819","wikidata":"https://www.wikidata.org/wiki/Q62542","display_name":"Phase space","level":2,"score":0.28790000081062317},{"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/C2779808786","wikidata":"https://www.wikidata.org/wiki/Q6664603","display_name":"Locality","level":2,"score":0.27630001306533813},{"id":"https://openalex.org/C2778926657","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum system","level":3,"score":0.27000001072883606},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.26739999651908875},{"id":"https://openalex.org/C2778755073","wikidata":"https://www.wikidata.org/wiki/Q10858537","display_name":"Scale (ratio)","level":2,"score":0.267300009727478},{"id":"https://openalex.org/C111030470","wikidata":"https://www.wikidata.org/wiki/Q1430460","display_name":"Curse of dimensionality","level":2,"score":0.26649999618530273},{"id":"https://openalex.org/C15706264","wikidata":"https://www.wikidata.org/wiki/Q230883","display_name":"Quantum state","level":3,"score":0.2653999924659729},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.260699987411499},{"id":"https://openalex.org/C121040770","wikidata":"https://www.wikidata.org/wiki/Q215675","display_name":"Quantum entanglement","level":3,"score":0.2605000138282776},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.25859999656677246},{"id":"https://openalex.org/C2781238097","wikidata":"https://www.wikidata.org/wiki/Q175026","display_name":"Object (grammar)","level":2,"score":0.25850000977516174},{"id":"https://openalex.org/C2781181686","wikidata":"https://www.wikidata.org/wiki/Q4226068","display_name":"Coherence (philosophical gambling strategy)","level":2,"score":0.2572000026702881},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.2549000084400177},{"id":"https://openalex.org/C155512373","wikidata":"https://www.wikidata.org/wiki/Q287450","display_name":"Residual","level":2,"score":0.25}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2609.33777","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.33777","pdf_url":"https://arxiv.org/pdf/2609.33777","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.2609.33777","doi":"https://doi.org/10.48550/arxiv.2609.33777","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2609.33777","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:2609.33777","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.33777","pdf_url":"https://arxiv.org/pdf/2609.33777","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":[],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7214722356.pdf","grobid_xml":"https://content.openalex.org/works/W7214722356.grobid-xml"},"referenced_works_count":15,"referenced_works":["https://openalex.org/W2016440527","https://openalex.org/W2267743026","https://openalex.org/W2760768235","https://openalex.org/W3205146778","https://openalex.org/W4226248670","https://openalex.org/W4296933636","https://openalex.org/W4407946832","https://openalex.org/W4408291274","https://openalex.org/W4410828900","https://openalex.org/W4410895338","https://openalex.org/W4412990627","https://openalex.org/W4417088643","https://openalex.org/W4417447048","https://openalex.org/W7133508828","https://openalex.org/W7203640659"],"related_works":["https://openalex.org/W4318719357","https://openalex.org/W4402273090","https://openalex.org/W4406386057","https://openalex.org/W4388685701","https://openalex.org/W7214739086","https://openalex.org/W4400416456","https://openalex.org/W4415798142","https://openalex.org/W7160267356","https://openalex.org/W6979227762","https://openalex.org/W4382407864"],"abstract_inverted_index":{"Phase":[0],"diagrams":[1],"and":[2,72,75,170,179],"energy":[3],"surfaces":[4],"require":[5],"low-energy":[6,42,63],"states":[7],"across":[8],"a":[9,28,47,92,121,151,157,191],"family":[10],"of":[11,50,144,194],"Hamiltonians,":[12],"but":[13,36],"solving":[14],"each":[15],"parameter":[16,34,163],"point":[17],"independently":[18],"is":[19,25,44,60,115],"prohibitively":[20],"expensive.":[21],"An":[22],"important":[23],"question":[24],"not":[26],"how":[27,37],"single":[29],"labeled":[30],"eigenstate":[31],"behaves":[32],"under":[33],"variation,":[35],"much":[38],"information":[39,146],"about":[40],"the":[41,56,61,96,116,137,142,175,180,186],"sector":[43],"contained":[45],"in":[46,86],"small":[48],"set":[49],"reference":[51],"wavefunctions.":[52,149],"We":[53],"show":[54],"that":[55,91],"physically":[57],"relevant":[58],"object":[59],"isolated":[62],"subspace":[64],"itself:":[65],"it":[66,76],"remains":[67],"well":[68],"defined":[69],"through":[70],"degeneracies":[71],"level":[73],"crossings,":[74],"can":[77],"be":[78],"learned":[79],"efficiently":[80],"from":[81,147],"nearby":[82],"sampled":[83,148],"states.":[84],"Working":[85],"this":[87,131,155],"subspace,":[88],"we":[89],"prove":[90],"sampling":[93,118,158],"pattern":[94],"cancelling":[95],"first":[97],"$q$":[98],"nonconstant":[99],"response":[100],"orders":[101],"yields":[102],"an":[103],"energy-density":[104],"error":[105],"bounded":[106],"by":[107],"$d^{2(q+1)}$":[108],"for":[109],"every":[110],"retained":[111],"level,":[112],"where":[113],"$d$":[114],"parameter-space":[117],"distance.":[119],"For":[120],"gapped":[122],"local":[123,152],"ground":[124],"state":[125],"on":[126,168],"$n$":[127],"sites,":[128],"locality":[129],"sharpens":[130],"to":[132],"$d^2(nd^2)^q$,":[133],"identifying":[134],"$nd^2$":[135,181],"as":[136],"natural":[138],"scaling":[139,177,187],"variable":[140],"governing":[141],"effectiveness":[143],"higher-order":[145],"In":[150],"analytic":[153],"regime":[154],"gives":[156],"cost":[159],"$K=\\mathcal{O}([\\log(1/\\varepsilon)]^D)$":[160],"at":[161],"fixed":[162],"dimension":[164],"$D$.":[165],"Numerical":[166],"results":[167],"Heisenberg":[169],"transverse-field":[171],"Ising":[172],"chains":[173],"confirm":[174],"distance":[176],"law":[178],"collapse.":[182],"Beyond":[183],"characterizing":[184],"learnability,":[185],"behavior":[188],"further":[189],"provides":[190],"finite-size":[192],"probe":[193],"phase":[195],"transitions":[196],"near":[197],"criticality.":[198]},"counts_by_year":[],"updated_date":"2026-11-07T11:25:07.637180","created_date":"2026-09-30T00:00:00"}