{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2019:SJ2LTTWSWTX7QT2S3373YU32LX","short_pith_number":"pith:SJ2LTTWS","schema_version":"1.0","canonical_sha256":"9274b9ced2b4eff84f52deffbc537a5dd1d9b1ceb63d9ff7e486324233186596","source":{"kind":"arxiv","id":"1908.06076","version":3},"attestation_state":"computed","paper":{"title":"Number-Theoretic Characterizations of Some Restricted Clifford+T Circuits","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"quant-ph","authors_text":"Andrew N. Glaudell, Matthew Amy, Neil J. Ross","submitted_at":"2019-08-16T17:56:17Z","abstract_excerpt":"Kliuchnikov, Maslov, and Mosca proved in 2012 that a $2\\times 2$ unitary matrix $V$ can be exactly represented by a single-qubit Clifford+$T$ circuit if and only if the entries of $V$ belong to the ring $\\mathbb{Z}[1/\\sqrt{2},i]$. Later that year, Giles and Selinger showed that the same restriction applies to matrices that can be exactly represented by a multi-qubit Clifford+$T$ circuit. These number-theoretic characterizations shed new light upon the structure of Clifford+$T$ circuits and led to remarkable developments in the field of quantum compiling. In the present paper, we provide number"},"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":"1908.06076","kind":"arxiv","version":3},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"quant-ph","submitted_at":"2019-08-16T17:56:17Z","cross_cats_sorted":[],"title_canon_sha256":"dbddea89ba3d0454bdd80ee803c4728d9bca0e3be3336c6685197ec05d6db7a2","abstract_canon_sha256":"391176827d6712a3d893de7eae0734b763c200341f9e50a0079339980ac408f5"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T00:53:40.505268Z","signature_b64":"htDyLaPe429C8zkhTwDxqtv/fS4jle5iQ+ulP3x8CjGmFo4UlpeY5CZysrkrg1n4rL3WhpcqIb7wYdBQ1VpkCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9274b9ced2b4eff84f52deffbc537a5dd1d9b1ceb63d9ff7e486324233186596","last_reissued_at":"2026-07-05T00:53:40.504614Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T00:53:40.504614Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Number-Theoretic Characterizations of Some Restricted Clifford+T Circuits","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"quant-ph","authors_text":"Andrew N. Glaudell, Matthew Amy, Neil J. Ross","submitted_at":"2019-08-16T17:56:17Z","abstract_excerpt":"Kliuchnikov, Maslov, and Mosca proved in 2012 that a $2\\times 2$ unitary matrix $V$ can be exactly represented by a single-qubit Clifford+$T$ circuit if and only if the entries of $V$ belong to the ring $\\mathbb{Z}[1/\\sqrt{2},i]$. Later that year, Giles and Selinger showed that the same restriction applies to matrices that can be exactly represented by a multi-qubit Clifford+$T$ circuit. These number-theoretic characterizations shed new light upon the structure of Clifford+$T$ circuits and led to remarkable developments in the field of quantum compiling. In the present paper, we provide number"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1908.06076","kind":"arxiv","version":3},"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/1908.06076/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":"1908.06076","created_at":"2026-07-05T00:53:40.504694+00:00"},{"alias_kind":"arxiv_version","alias_value":"1908.06076v3","created_at":"2026-07-05T00:53:40.504694+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1908.06076","created_at":"2026-07-05T00:53:40.504694+00:00"},{"alias_kind":"pith_short_12","alias_value":"SJ2LTTWSWTX7","created_at":"2026-07-05T00:53:40.504694+00:00"},{"alias_kind":"pith_short_16","alias_value":"SJ2LTTWSWTX7QT2S","created_at":"2026-07-05T00:53:40.504694+00:00"},{"alias_kind":"pith_short_8","alias_value":"SJ2LTTWS","created_at":"2026-07-05T00:53:40.504694+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":0,"sample":[{"citing_arxiv_id":"2606.12480","citing_title":"Geometric Algebra Quantum Gate Decomposition","ref_index":17,"is_internal_anchor":false}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/SJ2LTTWSWTX7QT2S3373YU32LX","json":"https://pith.science/pith/SJ2LTTWSWTX7QT2S3373YU32LX.json","graph_json":"https://pith.science/api/pith-number/SJ2LTTWSWTX7QT2S3373YU32LX/graph.json","events_json":"https://pith.science/api/pith-number/SJ2LTTWSWTX7QT2S3373YU32LX/events.json","paper":"https://pith.science/paper/SJ2LTTWS"},"agent_actions":{"view_html":"https://pith.science/pith/SJ2LTTWSWTX7QT2S3373YU32LX","download_json":"https://pith.science/pith/SJ2LTTWSWTX7QT2S3373YU32LX.json","view_paper":"https://pith.science/paper/SJ2LTTWS","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1908.06076&json=true","fetch_graph":"https://pith.science/api/pith-number/SJ2LTTWSWTX7QT2S3373YU32LX/graph.json","fetch_events":"https://pith.science/api/pith-number/SJ2LTTWSWTX7QT2S3373YU32LX/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/SJ2LTTWSWTX7QT2S3373YU32LX/action/timestamp_anchor","attest_storage":"https://pith.science/pith/SJ2LTTWSWTX7QT2S3373YU32LX/action/storage_attestation","attest_author":"https://pith.science/pith/SJ2LTTWSWTX7QT2S3373YU32LX/action/author_attestation","sign_citation":"https://pith.science/pith/SJ2LTTWSWTX7QT2S3373YU32LX/action/citation_signature","submit_replication":"https://pith.science/pith/SJ2LTTWSWTX7QT2S3373YU32LX/action/replication_record"}},"created_at":"2026-07-05T00:53:40.504694+00:00","updated_at":"2026-07-05T00:53:40.504694+00:00"}