{"id":"https://openalex.org/W7215124174","doi":"https://doi.org/10.48550/arxiv.2609.40107","title":"Verification Complexity and Extension of Classical Shadows","display_name":"Verification Complexity and Extension of Classical Shadows","publication_year":2026,"publication_date":"2026-09-30","ids":{"openalex":"https://openalex.org/W7215124174","doi":"https://doi.org/10.48550/arxiv.2609.40107"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2609.40107","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.40107","pdf_url":"https://arxiv.org/pdf/2609.40107","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/2609.40107","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5081107553","display_name":"Georgios Karaiskos","orcid":"https://orcid.org/0009-0005-8230-6461","observed_orcids":["https://orcid.org/0009-0005-8230-6461"]},"institutions":[{"id":"https://openalex.org/I206945453","display_name":"Paderborn University","ror":"https://ror.org/058kzsd48","country_code":"DE","type":"education","lineage":["https://openalex.org/I206945453"]}],"countries":["DE"],"is_corresponding":false,"raw_author_name":"Karaiskos, Georgios","raw_affiliation_strings":["Department of Computer Science , Paderborn University , Germany"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Computer Science , Paderborn University , Germany","institution_ids":["https://openalex.org/I206945453"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5025285529","display_name":"Asad Raza","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I75951250","display_name":"Freie Universit\u00e4t Berlin","ror":"https://ror.org/046ak2485","country_code":"DE","type":"education","lineage":["https://openalex.org/I75951250"]}],"countries":["DE"],"is_corresponding":false,"raw_author_name":"Raza, Asad","raw_affiliation_strings":["Dahlem Center for Complex Quantum Systems , Freie Universit\u00e4t Berlin , 14195 Berlin , Germany"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Dahlem Center for Complex Quantum Systems , Freie Universit\u00e4t Berlin , 14195 Berlin , Germany","institution_ids":["https://openalex.org/I75951250"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5056653308","display_name":"Dorian Rudolph","orcid":"https://orcid.org/0000-0002-2440-7388","observed_orcids":["https://orcid.org/0000-0002-2440-7388"]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Rudolph, Dorian","raw_affiliation_strings":["Department of Computer Science and"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Computer Science and","institution_ids":[]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5046561636","display_name":"Dax Enshan Koh","orcid":"https://orcid.org/0000-0002-8968-591X","observed_orcids":["https://orcid.org/0000-0002-8968-591X"]},"institutions":[{"id":"https://openalex.org/I168639165","display_name":"Singapore Institute of Technology","ror":"https://ror.org/01v2c2791","country_code":"SG","type":"education","lineage":["https://openalex.org/I168639165"]}],"countries":["SG"],"is_corresponding":false,"raw_author_name":"Koh, Dax Enshan","raw_affiliation_strings":["Engineering Cluster , Singapore Institute of Technology , 1 Punggol Coast Road , Singapore 828608 , Republic of Singapore ; Quantum Innovation Centre (Q.InC) ,"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Engineering Cluster , Singapore Institute of Technology , 1 Punggol Coast Road , Singapore 828608 , Republic of Singapore ; Quantum Innovation Centre (Q.InC) ,","institution_ids":["https://openalex.org/I168639165"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5023060969","display_name":"Sevag Gharibian","orcid":"https://orcid.org/0000-0002-9992-3379","observed_orcids":["https://orcid.org/0000-0002-9992-3379"]},"institutions":[{"id":"https://openalex.org/I206945453","display_name":"Paderborn University","ror":"https://ror.org/058kzsd48","country_code":"DE","type":"education","lineage":["https://openalex.org/I206945453"]}],"countries":["DE"],"is_corresponding":false,"raw_author_name":"Gharibian, Sevag","raw_affiliation_strings":["Department of Computer Science and","Institute for Photonic Quantum Systems (PhoQS) , Paderborn University , Germany"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Computer Science and","institution_ids":[]},{"raw_affiliation_string":"Institute for Photonic Quantum Systems (PhoQS) , Paderborn University , Germany","institution_ids":["https://openalex.org/I206945453"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":3,"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/T10682","display_name":"Quantum Computing Algorithms and Architecture","score":0.951200008392334,"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":0.951200008392334,"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/T10020","display_name":"Quantum Information and Cryptography","score":0.02889999933540821,"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/T11804","display_name":"Quantum many-body systems","score":0.010700000450015068,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/classical-shadows","display_name":"classical shadows","score":0.9679999947547913},{"id":"https://openalex.org/keywords/verification-complexity","display_name":"verification complexity","score":0.6899999976158142},{"id":"https://openalex.org/keywords/conp-hardness","display_name":"coNP-hardness","score":0.968999981880188},{"id":"https://openalex.org/keywords/pauli-strings","display_name":"Pauli strings","score":0.6209999918937683},{"id":"https://openalex.org/keywords/quantum-state-tomography","display_name":"quantum state tomography","score":0.5920000076293945}],"concepts":[{"id":"https://openalex.org/C32848918","wikidata":"https://www.wikidata.org/wiki/Q845789","display_name":"Observable","level":2,"score":0.550000011920929},{"id":"https://openalex.org/C48103436","wikidata":"https://www.wikidata.org/wiki/Q599031","display_name":"State (computer science)","level":2,"score":0.5336999893188477},{"id":"https://openalex.org/C84114770","wikidata":"https://www.wikidata.org/wiki/Q46344","display_name":"Quantum","level":2,"score":0.5188000202178955},{"id":"https://openalex.org/C2778029271","wikidata":"https://www.wikidata.org/wiki/Q5421931","display_name":"Extension (predicate logic)","level":2,"score":0.5037000179290771},{"id":"https://openalex.org/C31170391","wikidata":"https://www.wikidata.org/wiki/Q188619","display_name":"Hierarchy","level":2,"score":0.4941999912261963},{"id":"https://openalex.org/C117797892","wikidata":"https://www.wikidata.org/wiki/Q286363","display_name":"Shadow (psychology)","level":2,"score":0.44839999079704285},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.44760000705718994},{"id":"https://openalex.org/C15706264","wikidata":"https://www.wikidata.org/wiki/Q230883","display_name":"Quantum state","level":3,"score":0.41530001163482666},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.40560001134872437},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.36570000648498535},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.35580000281333923},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.3465999960899353},{"id":"https://openalex.org/C90119067","wikidata":"https://www.wikidata.org/wiki/Q43260","display_name":"Polynomial","level":2,"score":0.3458000123500824},{"id":"https://openalex.org/C80444323","wikidata":"https://www.wikidata.org/wiki/Q2878974","display_name":"Theoretical computer science","level":1,"score":0.33239999413490295},{"id":"https://openalex.org/C169699857","wikidata":"https://www.wikidata.org/wiki/Q2122243","display_name":"Quantum information","level":3,"score":0.3303000032901764},{"id":"https://openalex.org/C110340908","wikidata":"https://www.wikidata.org/wiki/Q131594","display_name":"Pauli exclusion principle","level":2,"score":0.2946000099182129},{"id":"https://openalex.org/C311688","wikidata":"https://www.wikidata.org/wiki/Q2393193","display_name":"Time complexity","level":2,"score":0.29269999265670776},{"id":"https://openalex.org/C180347388","wikidata":"https://www.wikidata.org/wiki/Q336233","display_name":"Pauli matrices","level":2,"score":0.2897000014781952},{"id":"https://openalex.org/C2776608160","wikidata":"https://www.wikidata.org/wiki/Q4785462","display_name":"Natural (archaeology)","level":2,"score":0.2712000012397766},{"id":"https://openalex.org/C2778926657","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum system","level":3,"score":0.27070000767707825},{"id":"https://openalex.org/C43682372","wikidata":"https://www.wikidata.org/wiki/Q7269080","display_name":"Quantum operation","level":4,"score":0.267300009727478},{"id":"https://openalex.org/C137002209","wikidata":"https://www.wikidata.org/wiki/Q898521","display_name":"Hidden variable theory","level":3,"score":0.263700008392334},{"id":"https://openalex.org/C2777299769","wikidata":"https://www.wikidata.org/wiki/Q3707858","display_name":"Type (biology)","level":2,"score":0.25699999928474426}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2609.40107","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.40107","pdf_url":"https://arxiv.org/pdf/2609.40107","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.2609.40107","doi":"https://doi.org/10.48550/arxiv.2609.40107","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2609.40107","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:2609.40107","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.40107","pdf_url":"https://arxiv.org/pdf/2609.40107","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/G1587758330","display_name":null,"funder_award_id":"SPP 2514","funder_id":"https://openalex.org/F4320320879","funder_display_name":"Deutsche Forschungsgemeinschaft"},{"id":"https://openalex.org/G1688234926","display_name":null,"funder_award_id":"CRC-183","funder_id":"https://openalex.org/F4320320879","funder_display_name":"Deutsche Forschungsgemeinschaft"},{"id":"https://openalex.org/G232088732","display_name":null,"funder_award_id":"S25Q9DA001","funder_id":"https://openalex.org/F4320320709","funder_display_name":"National Research Foundation Singapore"}],"funders":[{"id":"https://openalex.org/F4320320709","display_name":"National Research Foundation Singapore","ror":"https://ror.org/03cpyc314"},{"id":"https://openalex.org/F4320320879","display_name":"Deutsche Forschungsgemeinschaft","ror":"https://ror.org/018mejw64"},{"id":"https://openalex.org/F4320330828","display_name":"QuantERA","ror":null}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7215124174.pdf","grobid_xml":"https://content.openalex.org/works/W7215124174.grobid-xml"},"referenced_works_count":30,"referenced_works":["https://openalex.org/W17898807","https://openalex.org/W1765003530","https://openalex.org/W2016984055","https://openalex.org/W2047498859","https://openalex.org/W2101577710","https://openalex.org/W2131575873","https://openalex.org/W2165390647","https://openalex.org/W2195967161","https://openalex.org/W2405530305","https://openalex.org/W2611161794","https://openalex.org/W2767713609","https://openalex.org/W2793525843","https://openalex.org/W2805918291","https://openalex.org/W2880299821","https://openalex.org/W2946527621","https://openalex.org/W2963124215","https://openalex.org/W3006823398","https://openalex.org/W3099477842","https://openalex.org/W3107600208","https://openalex.org/W4293756338","https://openalex.org/W4298227433","https://openalex.org/W4310897979","https://openalex.org/W4388087330","https://openalex.org/W4394663479","https://openalex.org/W4402386416","https://openalex.org/W4404395958","https://openalex.org/W4408023888","https://openalex.org/W4409643501","https://openalex.org/W4410943977","https://openalex.org/W7211990334"],"related_works":["https://openalex.org/W4416385746","https://openalex.org/W4413904097","https://openalex.org/W4396987704","https://openalex.org/W4367854611","https://openalex.org/W4391679242","https://openalex.org/W4415183222","https://openalex.org/W4404346708","https://openalex.org/W4206245465","https://openalex.org/W4414741432","https://openalex.org/W7127081802"],"abstract_inverted_index":{"Classical":[0,61],"shadows":[1],"are":[2],"an":[3],"influential":[4],"framework":[5],"for":[6,111,115],"compressing":[7],"copies":[8],"of":[9,22,86,100,131,138,158,162,183,192],"a":[10,53,93,145,155,184],"given":[11],"quantum":[12,156,185],"state":[13,186,203,208],"$\u03c1$":[14,23,193,204],"into":[15],"classical":[16],"data":[17],"$S$,":[18,43],"enabling":[19],"many":[20],"properties":[21,85,99],"to":[24,82],"be":[25],"predicted":[26],"from":[27,52],"relatively":[28],"few":[29],"copies.":[30],"In":[31],"this":[32],"work,":[33],"we":[34,105,175],"study":[35],"two":[36],"natural":[37],"questions":[38],"involving":[39],"shadows:":[40],"(1)":[41],"Given":[42,77,178],"when":[44,126],"can":[45,88],"one":[46,81,89],"efficiently":[47,109],"verify":[48],"that":[49,79],"$S$":[50,78,182],"came":[51],"genuine":[54],"$n$-qubit":[55,133],"state?":[56,102],"This":[57],"is":[58,108,144,154,171,194,205],"called":[59],"the":[60,127,142,159,163,168,179,189,202,206],"Shadow":[62],"Validity":[63],"(CSV)":[64],"problem,":[65],"introduced":[66],"by":[67],"Karaiskos,":[68],"Rudolph,":[69],"Meyer,":[70],"Eisert,":[71],"and":[72,118,121,124],"Gharibian":[73],"[ICALP":[74],"2026].":[75],"(2)":[76],"allows":[80],"capture":[83],"2-local":[84,180],"$\u03c1$,":[87,187],"fake":[90],"or":[91],"spoof":[92],"shadow":[94],"$S'$":[95],"which":[96,167],"predicts":[97],"3-local":[98,190],"some":[101],"For":[103,173],"(1),":[104],"show":[106,176],"CSV":[107],"solvable":[110],"permutation-invariant":[112],"shadows,":[113,120],"QMA-hard":[114,125],"real,":[116],"fermionic,":[117],"bosonic":[119],"both":[122],"coNP-hard":[123],"observable":[128],"family":[129],"consists":[130],"all":[132],"Pauli":[134],"strings.":[135],"A":[136],"result":[137],"independent":[139],"interest":[140],"along":[141],"way":[143],"new":[146],"upper":[147],"bound":[148],"qc-$\u03a3_2$":[149,153],"$\\subseteq$":[150,198],"$\\mathrm{P}^{\\mathrm{PP}}$,":[151],"where":[152],"analogue":[157],"second":[160],"level":[161],"polynomial":[164],"hierarchy":[165],"in":[166],"first":[169],"proof":[170],"quantum.":[172],"(2),":[174],"intractability:":[177],"marginals":[181,191],"estimating":[188],"intractable":[195],"unless":[196],"QCMA":[197],"BPP,":[199],"even":[200],"if":[201],"unique":[207],"consistent":[209],"with":[210],"$S$.":[211]},"counts_by_year":[],"updated_date":"2026-11-07T11:25:07.637180","created_date":"2026-10-02T00:00:00"}
