{"paper":{"title":"Krylov Complexity","license":"http://creativecommons.org/licenses/by/4.0/","headline":"Krylov complexity measures operator growth in quantum systems without depending on arbitrary parameters.","cross_cats":["cond-mat.str-el","quant-ph"],"primary_cat":"hep-th","authors_text":"Adri\\'an S\\'anchez-Garrido, Eliezer Rabinovici, Julian Sonner, Ruth Shir","submitted_at":"2025-07-08T18:00:00Z","abstract_excerpt":"We introduce and review a new complexity measure, called `Krylov complexity', which takes its origins in the field of quantum-chaotic dynamics, serving as a canonical measure of operator growth and spreading. Krylov complexity, underpinned by the Lanczos algorithm, has since evolved into a highly diverse field of its own right, both because of its attractive features as a complexity, whose definition does not depend on arbitrary control parameters, and whose phenomenology serves as a rich and sensitive probe of chaotic dynamics up to exponentially late times, but also because of its relevance "},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"Krylov complexity is a canonical measure of operator growth and spreading whose definition does not depend on arbitrary control parameters and whose phenomenology serves as a rich and sensitive probe of chaotic dynamics up to exponentially late times.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"That the Lanczos algorithm yields a unique, physically meaningful complexity measure whose growth reliably distinguishes chaotic from integrable dynamics across all regimes, including holographic duals, without hidden parameter dependence.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Krylov complexity is a canonical, parameter-independent measure of operator spreading that probes chaotic dynamics to late times and admits a geometric interpretation in holographic duals.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Krylov complexity measures operator growth in quantum systems without depending on arbitrary parameters.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"687af415d35ea1192ccb905db1f542a5d9b1f1d2fa8decdd2bf5d2b79cd347f1"},"source":{"id":"2507.06286","kind":"arxiv","version":1},"verdict":{"id":"870b21a5-bd55-4048-8820-241077e69b30","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-16T14:48:57.960506Z","strongest_claim":"Krylov complexity is a canonical measure of operator growth and spreading whose definition does not depend on arbitrary control parameters and whose phenomenology serves as a rich and sensitive probe of chaotic dynamics up to exponentially late times.","one_line_summary":"Krylov complexity is a canonical, parameter-independent measure of operator spreading that probes chaotic dynamics to late times and admits a geometric interpretation in holographic duals.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"That the Lanczos algorithm yields a unique, physically meaningful complexity measure whose growth reliably distinguishes chaotic from integrable dynamics across all regimes, including holographic duals, without hidden parameter dependence.","pith_extraction_headline":"Krylov complexity measures operator growth in quantum systems without depending on arbitrary parameters."},"references":{"count":297,"sample":[{"doi":"","year":2013,"title":"author author Aaronson , Scott ( year 2013 ),\\ @noop title Quantum computing since Democritus \\ ( publisher Cambridge University Press ) NoStop","work_id":"2a301a6f-eb56-4a40-ba4c-652b22dcdb12","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"10.1002/prop.202400014","year":2024,"title":"author author Adhikari , Kiran , author Adwait \\ Rijal , author Ashok Kumar \\ Aryal , author Mausam \\ Ghimire , author Rajeev \\ Singh , and\\ author Christian \\ Deppe ( year 2024 ),\\ title title Krylov","work_id":"8a830c07-5e26-4f44-ba14-5eb32879238b","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"10.1088/1742-5468/ad0032","year":2023,"title":"author author Afrasiar , Mir , author Jaydeep \\ Kumar Basak , author Bidyut \\ Dey , author Kunal \\ Pal , and\\ author Kuntal \\ Pal ( year 2023 ),\\ title title Time evolution of spread complexity in que","work_id":"f03f0b85-9290-4401-9b1c-37d1ea359ea7","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"10.1007/jhep10(2024)107","year":2024,"title":"author author Aguilar-Gutierrez , Sergio E ( year 2024 ),\\ title title Towards complexity in de Sitter space from the doubled-scaled Sachdev-Ye-Kitaev model , \\ https://doi.org/10.1007/JHEP10(2024)107","work_id":"5e3d8bc7-50e2-414d-b8c9-2f92126781bb","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2025,"title":"Aguilar-Gutierrez,Building the Holographic Dictionary of the DSSYK from Chords, Complexity & Wormholes with Matter,2505.22716","work_id":"5387dedc-124e-4aa7-9179-09ddad4003f6","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":297,"snapshot_sha256":"628391b3a0d523763376986f05dcf08114bb662a33357d0eda66c1347d33dd1f","internal_anchors":44},"formal_canon":{"evidence_count":3,"snapshot_sha256":"2ba7636c2bca3f0e477d813765125aca1346e525ad00ffc5b81c51fb1c575ef0"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}