{"paper":{"title":"Polylogarithmic-Weight Dicke States in QAC$^0$ and Arbitrary Symmetric States in QAC$^0_f$","license":"http://creativecommons.org/licenses/by/4.0/","headline":"Explicit constant-depth circuits prepare polylog-weight Dicke states using only Toffoli gates.","cross_cats":["cs.DS"],"primary_cat":"quant-ph","authors_text":"Lucas Gretta, Malvika Raj Joshi, Meghal Gupta","submitted_at":"2026-04-16T17:57:08Z","abstract_excerpt":"An $n$-qubit Dicke state of weight $k$, is the uniform superposition over all $n$-bit strings of Hamming weight $k$. Dicke states are central to quantum algorithms exhibiting speedups, such as Decoded Quantum Interferometry (Jordan et al., \\emph{Nature}, 2025). In the NISQ era, quantum hardware is constrained by both depth and locality, motivating the question of which global operations suffice to prepare such states. QAC$^0$, the quantum analogue of AC$^0$, minimally extends local $O(1)$-depth quantum circuits by allowing arbitrary-width Toffoli (reversible AND) gates.\n  We show that Dicke st"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We give explicit constant-depth circuits that prepare n-qubit Dicke states for all k ≤ polylog(n), using only multi-qubit Toffoli gates and single-qubit unitaries. This gives the first QAC^0 construction of super-constant weight Dicke states.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The constructions assume the standard quantum circuit model with unbounded fan-in Toffoli gates allowed in constant depth and the prior definition of QAC^0 that permits only polylogarithmic fanout.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Explicit constant-depth circuits prepare super-constant weight Dicke states in QAC^0 and extend to arbitrary symmetric states with FANOUT_n.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Explicit constant-depth circuits prepare polylog-weight Dicke states using only Toffoli gates.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"1e8aed74fbe5f20206b8981dc12d349360d5883bbb886f56d004f8e1c25c734d"},"source":{"id":"2604.15298","kind":"arxiv","version":2},"verdict":{"id":"c9765937-74cd-4890-ba5e-d76211a9a94b","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-10T11:13:19.906873Z","strongest_claim":"We give explicit constant-depth circuits that prepare n-qubit Dicke states for all k ≤ polylog(n), using only multi-qubit Toffoli gates and single-qubit unitaries. This gives the first QAC^0 construction of super-constant weight Dicke states.","one_line_summary":"Explicit constant-depth circuits prepare super-constant weight Dicke states in QAC^0 and extend to arbitrary symmetric states with FANOUT_n.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The constructions assume the standard quantum circuit model with unbounded fan-in Toffoli gates allowed in constant depth and the prior definition of QAC^0 that permits only polylogarithmic fanout.","pith_extraction_headline":"Explicit constant-depth circuits prepare polylog-weight Dicke states using only Toffoli gates."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2604.15298/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"}