{"paper":{"title":"Unitary designs in nearly optimal depth","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["cs.CC","cs.IT","math-ph","math.IT","math.MP"],"primary_cat":"quant-ph","authors_text":"Fernando Brandao, Hsin-Yuan Huang, Laura Cui, Thomas Schuster","submitted_at":"2025-07-08T17:48:33Z","abstract_excerpt":"We construct $\\varepsilon$-approximate unitary $k$-designs on $n$ qubits in circuit depth $O(\\log k \\log \\log n k / \\varepsilon)$. The depth is exponentially improved over all known results in all three parameters $n$, $k$, $\\varepsilon$. We further show that each dependence is optimal up to exponentially smaller factors. Our construction uses $\\tilde{{O}}(nk)$ ancilla qubits and ${O}(nk)$ bits of randomness, which are also optimal up to $\\log(n k)$ factors. An alternative construction achieves a smaller ancilla count $\\tilde{{O}}(n)$ with circuit depth ${O}(k \\log \\log nk/\\varepsilon)$. To ac"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2507.06216","kind":"arxiv","version":2},"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/2507.06216/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"}