{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2004:JSX5FU5FPWKHXFWRTP327UT5TV","short_pith_number":"pith:JSX5FU5F","schema_version":"1.0","canonical_sha256":"4cafd2d3a57d947b96d19bf7afd27d9d5fceeb09ad57caf42596bd3d97fd83d3","source":{"kind":"arxiv","id":"quant-ph/0407082","version":4},"attestation_state":"computed","paper":{"title":"Efficient Quantum Circuits for Schur and Clebsch-Gordan Transforms","license":"","headline":"","cross_cats":[],"primary_cat":"quant-ph","authors_text":"Aram Harrow, Dave Bacon, Isaac Chuang","submitted_at":"2004-07-12T20:09:01Z","abstract_excerpt":"The Schur basis on n d-dimensional quantum systems is a generalization of the total angular momentum basis that is useful for exploiting symmetry under permutations or collective unitary rotations. We present efficient (size poly(n,d,log(1/\\epsilon)) for accuracy \\epsilon) quantum circuits for the Schur transform, which is the change of basis between the computational and the Schur bases. These circuits are based on efficient circuits for the Clebsch-Gordan transformation. We also present an efficient circuit for a limited version of the Schur transform in which one needs only to project onto "},"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":false,"formal_links_present":false},"canonical_record":{"source":{"id":"quant-ph/0407082","kind":"arxiv","version":4},"metadata":{"license":"","primary_cat":"quant-ph","submitted_at":"2004-07-12T20:09:01Z","cross_cats_sorted":[],"title_canon_sha256":"78bb3f6713a55813720603cea3e659c4a64453c7c6d34c74911f00a9105a5ff4","abstract_canon_sha256":"206e0c1da84a96dae72170d1231a89544e3f7b3ebb81dd114e57ba8700074fcb"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-04T14:49:30.540900Z","signature_b64":"wNxljHWUYMTluWP/Rvk26AmcvJ5W6PfCnrF+DDI9a1rfuJEfqUcuxtzIv0ohtFcomXbG8FNt+wQGt3/OT9H9Aw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4cafd2d3a57d947b96d19bf7afd27d9d5fceeb09ad57caf42596bd3d97fd83d3","last_reissued_at":"2026-07-04T14:49:30.540524Z","signature_status":"signed_v1","first_computed_at":"2026-07-04T14:49:30.540524Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Efficient Quantum Circuits for Schur and Clebsch-Gordan Transforms","license":"","headline":"","cross_cats":[],"primary_cat":"quant-ph","authors_text":"Aram Harrow, Dave Bacon, Isaac Chuang","submitted_at":"2004-07-12T20:09:01Z","abstract_excerpt":"The Schur basis on n d-dimensional quantum systems is a generalization of the total angular momentum basis that is useful for exploiting symmetry under permutations or collective unitary rotations. We present efficient (size poly(n,d,log(1/\\epsilon)) for accuracy \\epsilon) quantum circuits for the Schur transform, which is the change of basis between the computational and the Schur bases. These circuits are based on efficient circuits for the Clebsch-Gordan transformation. We also present an efficient circuit for a limited version of the Schur transform in which one needs only to project onto "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"quant-ph/0407082","kind":"arxiv","version":4},"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/quant-ph/0407082/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"},"aliases":[{"alias_kind":"arxiv","alias_value":"quant-ph/0407082","created_at":"2026-07-04T14:49:30.540582+00:00"},{"alias_kind":"arxiv_version","alias_value":"quant-ph/0407082v4","created_at":"2026-07-04T14:49:30.540582+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.quant-ph/0407082","created_at":"2026-07-04T14:49:30.540582+00:00"},{"alias_kind":"pith_short_12","alias_value":"JSX5FU5FPWKH","created_at":"2026-07-04T14:49:30.540582+00:00"},{"alias_kind":"pith_short_16","alias_value":"JSX5FU5FPWKHXFWR","created_at":"2026-07-04T14:49:30.540582+00:00"},{"alias_kind":"pith_short_8","alias_value":"JSX5FU5F","created_at":"2026-07-04T14:49:30.540582+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":2,"internal_anchor_count":2,"sample":[{"citing_arxiv_id":"2605.15076","citing_title":"Deforming the Trail: Baseline Quantum Circuitry for $\\text{SU(2)}_k$ Lattice Gauge Theory","ref_index":43,"is_internal_anchor":true},{"citing_arxiv_id":"2405.09626","citing_title":"Permutation tests for quantum state identity","ref_index":1,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/JSX5FU5FPWKHXFWRTP327UT5TV","json":"https://pith.science/pith/JSX5FU5FPWKHXFWRTP327UT5TV.json","graph_json":"https://pith.science/api/pith-number/JSX5FU5FPWKHXFWRTP327UT5TV/graph.json","events_json":"https://pith.science/api/pith-number/JSX5FU5FPWKHXFWRTP327UT5TV/events.json","paper":"https://pith.science/paper/JSX5FU5F"},"agent_actions":{"view_html":"https://pith.science/pith/JSX5FU5FPWKHXFWRTP327UT5TV","download_json":"https://pith.science/pith/JSX5FU5FPWKHXFWRTP327UT5TV.json","view_paper":"https://pith.science/paper/JSX5FU5F","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=quant-ph/0407082&json=true","fetch_graph":"https://pith.science/api/pith-number/JSX5FU5FPWKHXFWRTP327UT5TV/graph.json","fetch_events":"https://pith.science/api/pith-number/JSX5FU5FPWKHXFWRTP327UT5TV/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/JSX5FU5FPWKHXFWRTP327UT5TV/action/timestamp_anchor","attest_storage":"https://pith.science/pith/JSX5FU5FPWKHXFWRTP327UT5TV/action/storage_attestation","attest_author":"https://pith.science/pith/JSX5FU5FPWKHXFWRTP327UT5TV/action/author_attestation","sign_citation":"https://pith.science/pith/JSX5FU5FPWKHXFWRTP327UT5TV/action/citation_signature","submit_replication":"https://pith.science/pith/JSX5FU5FPWKHXFWRTP327UT5TV/action/replication_record"}},"created_at":"2026-07-04T14:49:30.540582+00:00","updated_at":"2026-07-04T14:49:30.540582+00:00"}