{"id":"https://openalex.org/W7215117666","doi":"https://doi.org/10.48550/arxiv.2609.40278","title":"Computational Bounds for $f$-Routing","display_name":"Computational Bounds for $f$-Routing","publication_year":2026,"publication_date":"2026-09-30","ids":{"openalex":"https://openalex.org/W7215117666","doi":"https://doi.org/10.48550/arxiv.2609.40278"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2609.40278","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.40278","pdf_url":"https://arxiv.org/pdf/2609.40278","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/2609.40278","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5014429949","display_name":"Oren Renard","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I205783295","display_name":"Cornell University","ror":"https://ror.org/05bnh6r87","country_code":"US","type":"education","lineage":["https://openalex.org/I205783295"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Renard, Oren","raw_affiliation_strings":["Cornell University"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Cornell University","institution_ids":["https://openalex.org/I205783295"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5072056838","display_name":"Nicholas Spooner","orcid":"https://orcid.org/0000-0002-0085-2137","observed_orcids":["https://orcid.org/0000-0002-0085-2137"]},"institutions":[{"id":"https://openalex.org/I205783295","display_name":"Cornell University","ror":"https://ror.org/05bnh6r87","country_code":"US","type":"education","lineage":["https://openalex.org/I205783295"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Spooner, Nicholas","raw_affiliation_strings":["Cornell University"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Cornell University","institution_ids":["https://openalex.org/I205783295"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":[],"corresponding_institution_ids":["https://openalex.org/I205783295"],"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/T10020","display_name":"Quantum Information and Cryptography","score":0.9794999957084656,"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/T10020","display_name":"Quantum Information and Cryptography","score":0.9794999957084656,"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":0.01730000041425228,"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/T10237","display_name":"Cryptography and Data Security","score":0.0012000000569969416,"subfield":{"id":"https://openalex.org/subfields/1705","display_name":"Computer Networks and Communications"},"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/clifford-t-circuits","display_name":"Clifford+T circuits","score":0.9369999766349792},{"id":"https://openalex.org/keywords/time-hierarchy","display_name":"time hierarchy","score":0.6119999885559082},{"id":"https://openalex.org/keywords/spatial-hierarchy","display_name":"spatial hierarchy","score":0.6869999766349792}],"concepts":[{"id":"https://openalex.org/C203087015","wikidata":"https://www.wikidata.org/wiki/Q378201","display_name":"Qubit","level":3,"score":0.5986999869346619},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.560699999332428},{"id":"https://openalex.org/C58053490","wikidata":"https://www.wikidata.org/wiki/Q176555","display_name":"Quantum computer","level":3,"score":0.5335000157356262},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.5218999981880188},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.49889999628067017},{"id":"https://openalex.org/C39637292","wikidata":"https://www.wikidata.org/wiki/Q908207","display_name":"Complexity class","level":3,"score":0.4855000078678131},{"id":"https://openalex.org/C311688","wikidata":"https://www.wikidata.org/wiki/Q2393193","display_name":"Time complexity","level":2,"score":0.45190000534057617},{"id":"https://openalex.org/C108710211","wikidata":"https://www.wikidata.org/wiki/Q11538","display_name":"Mathematical proof","level":2,"score":0.42500001192092896},{"id":"https://openalex.org/C159718280","wikidata":"https://www.wikidata.org/wiki/Q5526353","display_name":"Gas meter prover","level":3,"score":0.41370001435279846},{"id":"https://openalex.org/C2078106","wikidata":"https://www.wikidata.org/wiki/Q14906620","display_name":"Accumulator (cryptography)","level":2,"score":0.41110000014305115},{"id":"https://openalex.org/C2779014939","wikidata":"https://www.wikidata.org/wiki/Q6510239","display_name":"Learning with errors","level":3,"score":0.3986000120639801},{"id":"https://openalex.org/C84114770","wikidata":"https://www.wikidata.org/wiki/Q46344","display_name":"Quantum","level":2,"score":0.3935000002384186},{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.3806999921798706},{"id":"https://openalex.org/C45374587","wikidata":"https://www.wikidata.org/wiki/Q12525525","display_name":"Computation","level":2,"score":0.37790000438690186},{"id":"https://openalex.org/C137019171","wikidata":"https://www.wikidata.org/wiki/Q2623817","display_name":"Quantum algorithm","level":3,"score":0.3776000142097473},{"id":"https://openalex.org/C179799912","wikidata":"https://www.wikidata.org/wiki/Q205084","display_name":"Computational complexity theory","level":2,"score":0.37709999084472656},{"id":"https://openalex.org/C91399829","wikidata":"https://www.wikidata.org/wiki/Q1115684","display_name":"Commitment scheme","level":3,"score":0.35920000076293945},{"id":"https://openalex.org/C33676613","wikidata":"https://www.wikidata.org/wiki/Q13415176","display_name":"Dimension (graph theory)","level":2,"score":0.3497999906539917},{"id":"https://openalex.org/C2777212361","wikidata":"https://www.wikidata.org/wiki/Q5127848","display_name":"Class (philosophy)","level":2,"score":0.34880000352859497},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.33820000290870667},{"id":"https://openalex.org/C90673727","wikidata":"https://www.wikidata.org/wiki/Q901718","display_name":"Product (mathematics)","level":2,"score":0.336899995803833},{"id":"https://openalex.org/C187455244","wikidata":"https://www.wikidata.org/wiki/Q942353","display_name":"Boolean function","level":2,"score":0.33469998836517334},{"id":"https://openalex.org/C2777027219","wikidata":"https://www.wikidata.org/wiki/Q1284190","display_name":"Constant (computer programming)","level":2,"score":0.33160001039505005},{"id":"https://openalex.org/C80444323","wikidata":"https://www.wikidata.org/wiki/Q2878974","display_name":"Theoretical computer science","level":1,"score":0.3244999945163727},{"id":"https://openalex.org/C58948655","wikidata":"https://www.wikidata.org/wiki/Q7604392","display_name":"Statistical distance","level":3,"score":0.3174000084400177},{"id":"https://openalex.org/C90119067","wikidata":"https://www.wikidata.org/wiki/Q43260","display_name":"Polynomial","level":2,"score":0.31700000166893005},{"id":"https://openalex.org/C198082294","wikidata":"https://www.wikidata.org/wiki/Q3399648","display_name":"Position (finance)","level":2,"score":0.31049999594688416},{"id":"https://openalex.org/C193319292","wikidata":"https://www.wikidata.org/wiki/Q272172","display_name":"Hamming distance","level":2,"score":0.30809998512268066},{"id":"https://openalex.org/C2777704519","wikidata":"https://www.wikidata.org/wiki/Q45732","display_name":"MAGIC (telescope)","level":2,"score":0.2906999886035919},{"id":"https://openalex.org/C178489894","wikidata":"https://www.wikidata.org/wiki/Q8789","display_name":"Cryptography","level":2,"score":0.28459998965263367},{"id":"https://openalex.org/C176329583","wikidata":"https://www.wikidata.org/wiki/Q191943","display_name":"Zero-knowledge proof","level":3,"score":0.2824000120162964},{"id":"https://openalex.org/C92043244","wikidata":"https://www.wikidata.org/wiki/Q7269023","display_name":"Quantum complexity theory","level":4,"score":0.2777000069618225},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.266400009393692},{"id":"https://openalex.org/C2776711565","wikidata":"https://www.wikidata.org/wiki/Q7445058","display_name":"Security parameter","level":3,"score":0.2662000060081482},{"id":"https://openalex.org/C184720557","wikidata":"https://www.wikidata.org/wiki/Q7825049","display_name":"Topology (electrical circuits)","level":2,"score":0.25679999589920044},{"id":"https://openalex.org/C127964446","wikidata":"https://www.wikidata.org/wiki/Q1092142","display_name":"Computational resource","level":3,"score":0.2563999891281128},{"id":"https://openalex.org/C55689738","wikidata":"https://www.wikidata.org/wiki/Q15963867","display_name":"Discrete time and continuous time","level":2,"score":0.2506999969482422},{"id":"https://openalex.org/C97399411","wikidata":"https://www.wikidata.org/wiki/Q825367","display_name":"Coin flipping","level":2,"score":0.25049999356269836}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2609.40278","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.40278","pdf_url":"https://arxiv.org/pdf/2609.40278","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.2609.40278","doi":"https://doi.org/10.48550/arxiv.2609.40278","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2609.40278","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:2609.40278","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.40278","pdf_url":"https://arxiv.org/pdf/2609.40278","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/W7215117666.pdf","grobid_xml":"https://content.openalex.org/works/W7215117666.grobid-xml"},"referenced_works_count":33,"referenced_works":["https://openalex.org/W228759579","https://openalex.org/W1543653153","https://openalex.org/W1585946329","https://openalex.org/W1840167064","https://openalex.org/W1999502904","https://openalex.org/W2045436532","https://openalex.org/W2052146120","https://openalex.org/W2071584320","https://openalex.org/W2120279343","https://openalex.org/W2159363909","https://openalex.org/W2165230677","https://openalex.org/W2165342591","https://openalex.org/W2199001409","https://openalex.org/W2245439539","https://openalex.org/W2474499318","https://openalex.org/W2741846989","https://openalex.org/W2761012467","https://openalex.org/W2804172638","https://openalex.org/W2805649731","https://openalex.org/W2990961515","https://openalex.org/W3134161675","https://openalex.org/W4287214698","https://openalex.org/W4293240600","https://openalex.org/W4310117831","https://openalex.org/W4385708385","https://openalex.org/W4394660191","https://openalex.org/W4400089995","https://openalex.org/W4406662110","https://openalex.org/W4415264086","https://openalex.org/W7110046292","https://openalex.org/W7110068678","https://openalex.org/W7148379860","https://openalex.org/W7160320838"],"related_works":["https://openalex.org/W4385708385","https://openalex.org/W4406662110","https://openalex.org/W7214040677","https://openalex.org/W3156399423","https://openalex.org/W4415098718","https://openalex.org/W7203720774","https://openalex.org/W4221138685","https://openalex.org/W3147957109","https://openalex.org/W1566874944","https://openalex.org/W2047671131"],"abstract_inverted_index":{"The":[0],"$f$-routing":[1],"protocol":[2],"is":[3,105],"a":[4,63,68],"leading":[5],"candidate":[6],"for":[7,22,33,58,207],"quantum":[8,98,111],"position":[9],"verification":[10],"(Kent,":[11],"Munro,":[12],"and":[13,62,75,123,159,171,181,202,220],"Spiller,":[14],"Physical":[15],"Review":[16],"A,":[17],"2011),":[18],"but":[19],"security":[20],"guarantees":[21],"explicit":[23,150,176],"functions":[24,177],"remain":[25],"limited.":[26],"We":[27,55],"prove":[28],"unconditional":[29],"resource":[30],"lower":[31],"bounds":[32,40,89,191],"uniform":[34,69],"attackers.":[35],"Our":[36],"techniques":[37],"bypass":[38],"communication-complexity":[39],"central":[41],"to":[42],"previous":[43],"works,":[44],"which":[45,116],"are":[46,96,133,149],"inherently":[47],"at":[48,79,137,155,161,213],"most":[49,138,156,162],"linear":[50],"in":[51,72,115,120,126],"the":[52,86,94,106,117,124,131,147,193,208,214],"input":[53,59,84],"length.":[54],"show":[56],"that,":[57],"length":[60],"$n$":[61],"sufficiently":[64],"small":[65],"constant":[66],"$\u03b5>0$,":[67],"strategy":[70],"running":[71],"time":[73,121,127],"$T$":[74,122,157],"succeeding":[76],"with":[77],"probability":[78],"least":[80],"$1-\u03b5$":[81],"on":[82,90,154],"every":[83],"implies":[85],"following":[87],"computational":[88,188],"$f$.":[91],"1.":[92],"If":[93,130,146],"strategies":[95,132,148],"arbitrary":[97],"channels,":[99],"then":[100,142,167,174],"$f\\in":[101,143,168],"\\mathrm{QSZK}(\\mathrm{poly}(T))$,":[102],"where":[103],"$\\mathrm{QSZK}(T)$":[104],"class":[107],"of":[108,199,216],"languages":[109],"having":[110,160],"statistical":[112],"zero-knowledge":[113],"proofs":[114],"verifier":[118],"runs":[119],"simulator":[125],"$\\mathrm{poly}(T)$.":[128],"2.":[129],"Clifford+T":[134],"circuits":[135],"using":[136],"$t$":[139],"magic":[140],"gates,":[141],"\\mathrm{DTIME}(\\mathrm{poly}(T2^t))$.":[144],"3.":[145],"Pauli-sparse":[151],"unitaries":[152],"acting":[153],"qubits":[158],"$s$":[163],"nonzero":[164],"Pauli":[165],"coefficients,":[166],"\\mathrm{DTIME}(\\mathrm{poly}(Ts))$.":[169],"Time":[170],"space":[172],"hierarchies":[173],"yield":[175],"secure":[178],"against":[179],"superlinear":[180],"even":[182],"all":[183],"polynomial-time":[184],"adversaries":[185],"under":[186],"our":[187],"restrictions.":[189],"These":[190],"exceed":[192],"bounded-storage":[194],"qubit":[195],"bound":[196],"$q\\leq\\log":[197],"n$":[198],"Bluhm,":[200],"Christandl,":[201],"Speelman":[203],"(Nature":[204],"Physics,":[205],"2022)":[206],"inner":[209],"product":[210],"function":[211],"$f=\\mathrm{IP}$,":[212],"cost":[215],"restricting":[217],"adversarial":[218],"computation":[219],"increasing":[221],"honest":[222],"evaluation":[223],"complexity.":[224]},"counts_by_year":[],"updated_date":"2026-11-08T12:12:28.789561","created_date":"2026-10-02T00:00:00"}