{"paper":{"title":"Polynomial Degree and Lower Bounds in Quantum Complexity: Collision and Element Distinctness with Small Range","license":"","headline":"","cross_cats":["cs.CC"],"primary_cat":"quant-ph","authors_text":"Andris Ambainis","submitted_at":"2003-05-29T16:20:35Z","abstract_excerpt":"We give a general method for proving quantum lower bounds for problems with small range. Namely, we show that, for any symmetric problem defined on functions $f:\\{1, ..., N\\}\\to\\{1, ..., M\\}$, its polynomial degree is the same for all $M\\geq N$. Therefore, if we have a quantum lower bound for some (possibly, quite large) range $M$ which is shown using polynomials method, we immediately get the same lower bound for all ranges $M\\geq N$. In particular, we get $\\Omega(N^{1/3})$ and $\\Omega(N^{2/3})$ quantum lower bounds for collision and element distinctness with small range."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"quant-ph/0305179","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/quant-ph/0305179/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}