{"paper":{"title":"Explicit orthogonal and unitary designs","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"cs.CC","authors_text":"Pedro Paredes, Rocco A. Servedio, Ryan O'Donnell","submitted_at":"2023-10-20T15:46:15Z","abstract_excerpt":"We give a strongly explicit construction of $\\varepsilon$-approximate $k$-designs for the orthogonal group $\\mathrm{O}(N)$ and the unitary group $\\mathrm{U}(N)$, for $N=2^n$. Our designs are of cardinality $\\mathrm{poly}(N^k/\\varepsilon)$ (equivalently, they have seed length $O(nk + \\log(1/\\varepsilon)))$; up to the polynomial, this matches the number of design elements used by the construction consisting of completely random matrices."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2310.13597","kind":"arxiv","version":1},"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/2310.13597/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"}