{"paper":{"title":"Query and Depth Upper Bounds for Quantum Unitaries via Grover Search","license":"http://creativecommons.org/licenses/by/4.0/","headline":"Any n-qubit unitary can be implemented approximately in time Õ(2^{n/2}) with oracle queries or exactly in circuit depth Õ(2^{n/2}) with ancillae.","cross_cats":["cs.CC"],"primary_cat":"quant-ph","authors_text":"Gregory Rosenthal","submitted_at":"2021-11-15T18:53:48Z","abstract_excerpt":"We prove that any $n$-qubit unitary can be implemented (i) approximately in time $\\tilde O\\big(2^{n/2}\\big)$ with query access to an appropriate classical oracle, and also (ii) exactly by a circuit of depth $\\tilde O\\big(2^{n/2}\\big)$ with one- and two-qubit gates and $2^{O(n)}$ ancillae. The proofs involve similar reductions to Grover search. The proof of (ii) also involves a linear-depth construction of arbitrary quantum states using one- and two-qubit gates (in fact, this can be improved to constant depth with the addition of fanout and generalized Toffoli gates) which may be of independent"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We prove that any n-qubit unitary can be implemented (i) approximately in time Õ(2^{n/2}) with query access to an appropriate classical oracle, and also (ii) exactly by a circuit of depth Õ(2^{n/2}) with one- and two-qubit gates and 2^{O(n)} ancillae. The proofs involve similar reductions to Grover search.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The existence and query accessibility of an 'appropriate classical oracle' encoding the unitary for part (i), together with the availability of 2^{O(n)} ancilla qubits and the linear-depth arbitrary state preparation subroutine for part (ii); if these modeling assumptions fail or the reductions do not hold, the stated bounds are invalidated. This premise enters directly in the abstract's statement of the two main results.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Any n-qubit unitary can be implemented approximately with Õ(2^{n/2}) oracle queries or exactly with Õ(2^{n/2}) circuit depth via Grover search reductions, with matching lower bounds for certain implementations.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Any n-qubit unitary can be implemented approximately in time Õ(2^{n/2}) with oracle queries or exactly in circuit depth Õ(2^{n/2}) with ancillae.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"64c757847b44b3e58ea3c69f71008bc9232551e4c2f40846a052d79155527a97"},"source":{"id":"2111.07992","kind":"arxiv","version":5},"verdict":{"id":"ae2b4a00-fa69-47e6-991a-58fb7fe4b2b9","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-24T13:23:19.677434Z","strongest_claim":"We prove that any n-qubit unitary can be implemented (i) approximately in time Õ(2^{n/2}) with query access to an appropriate classical oracle, and also (ii) exactly by a circuit of depth Õ(2^{n/2}) with one- and two-qubit gates and 2^{O(n)} ancillae. The proofs involve similar reductions to Grover search.","one_line_summary":"Any n-qubit unitary can be implemented approximately with Õ(2^{n/2}) oracle queries or exactly with Õ(2^{n/2}) circuit depth via Grover search reductions, with matching lower bounds for certain implementations.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The existence and query accessibility of an 'appropriate classical oracle' encoding the unitary for part (i), together with the availability of 2^{O(n)} ancilla qubits and the linear-depth arbitrary state preparation subroutine for part (ii); if these modeling assumptions fail or the reductions do not hold, the stated bounds are invalidated. This premise enters directly in the abstract's statement of the two main results.","pith_extraction_headline":"Any n-qubit unitary can be implemented approximately in time Õ(2^{n/2}) with oracle queries or exactly in circuit depth Õ(2^{n/2}) with ancillae."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2111.07992/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":24,"sample":[{"doi":"","year":2021,"title":"Open problems related to quantum query complexity","work_id":"1c1b055e-cfb6-4172-aeca-fdb99192ddc6","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2016,"title":"The Complexity of Quantum States and Transformations: From Quantum Money to Black Holes","work_id":"d0452c4e-3afd-4f24-9ebc-33a4213cace1","ref_index":2,"cited_arxiv_id":"1607.05256","is_internal_anchor":true},{"doi":"10.4086/toc.2007.v003a007","year":2007,"title":"Quantum versus classical proofs and advice","work_id":"e48d873d-1f2d-49f3-a166-1e3f1cff2717","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"10.1006/jcss.2002.1826","year":2002,"title":"Quantum lower bounds by quantum arguments","work_id":"55a1d840-a21e-45c1-a34e-2ec8ab7f1cf4","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"10.1090/conm/305/05215","year":2002,"title":"Quantum Amplitude Amplification and Estimation","work_id":"c8922d48-1de0-498c-8b54-887d90f4834c","ref_index":5,"cited_arxiv_id":"quant-ph/0005055","is_internal_anchor":true}],"resolved_work":24,"snapshot_sha256":"0d1c057efcf58d861c60399f89c324c6c22e4e7e7761b0df80ab2a88bb3bb371","internal_anchors":7},"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"}