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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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":true,"formal_links_present":false},"canonical_record":{"source":{"id":"2111.07992","kind":"arxiv","version":5},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"quant-ph","submitted_at":"2021-11-15T18:53:48Z","cross_cats_sorted":["cs.CC"],"title_canon_sha256":"9aeed3136e1d352ab52bf4ba45341286f245e436b06bc6811ef60edaf2feecaa","abstract_canon_sha256":"ae24996a0fb32001784784d0c4a899dc18a5f4d72ec29096829f2d1b54eb9b59"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-01T14:23:04.605739Z","signature_b64":"bmpz3IsH7OH+a4gsdbmtRSAlaBBrdmXQ05w0g91kveXRv19FSmgVYyFrSw2Hjwi33kfDKatsFfiDQEvyGgJkDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b919d58efbe33966c56821c303bc601cd2d9fa95f2cf80b57c8a0baf98c224fa","last_reissued_at":"2026-07-01T14:23:04.605215Z","signature_status":"signed_v1","first_computed_at":"2026-07-01T14:23:04.605215Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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. 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