{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2022:RTIAM7FNL6ZHHICWMDPG2HANKC","short_pith_number":"pith:RTIAM7FN","schema_version":"1.0","canonical_sha256":"8cd0067cad5fb273a05660de6d1c0d50aa380050ec8aea87531ac2bb72d571f3","source":{"kind":"arxiv","id":"2204.13176","version":1},"attestation_state":"computed","paper":{"title":"Divisible Codes for Quantum Computation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.IT","math.IT"],"primary_cat":"quant-ph","authors_text":"Jingzhen Hu, Qingzhong Liang, Robert Calderbank","submitted_at":"2022-04-27T20:18:51Z","abstract_excerpt":"Divisible codes are defined by the property that codeword weights share a common divisor greater than one. They are used to design signals for communications and sensing, and this paper explores how they can be used to protect quantum information as it is transformed by logical gates. Given a CSS code $\\mathcal{C}$, we derive conditions that are both necessary and sufficient for a transversal diagonal physical operator $U_Z$ to preserve $\\mathcal{C}$ and induce $U_L$. The group of $Z$-stabilizers in a CSS code $\\mathcal{C}$ is determined by the dual of a classical $[n, k_1]$ binary code $\\math"},"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":"2204.13176","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"quant-ph","submitted_at":"2022-04-27T20:18:51Z","cross_cats_sorted":["cs.IT","math.IT"],"title_canon_sha256":"b833deae85ff5263d5cca9bcc9796ad8da45b1d19fa85a7b16304019ed7f1f53","abstract_canon_sha256":"b4d42685a1c1a6b3efa9b23665a39484c7ae3c7720352a5ea51450bc95250cb7"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T04:18:31.740710Z","signature_b64":"MoJyjU/umq8uB24eJc9t5EbSeD8eMXXwtPhB/uUhsM0AsD0T9yMdhW3StMmciiPxscmSHEOAEJTOuCkQDdqMDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8cd0067cad5fb273a05660de6d1c0d50aa380050ec8aea87531ac2bb72d571f3","last_reissued_at":"2026-07-05T04:18:31.740299Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T04:18:31.740299Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Divisible Codes for Quantum Computation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.IT","math.IT"],"primary_cat":"quant-ph","authors_text":"Jingzhen Hu, Qingzhong Liang, Robert Calderbank","submitted_at":"2022-04-27T20:18:51Z","abstract_excerpt":"Divisible codes are defined by the property that codeword weights share a common divisor greater than one. They are used to design signals for communications and sensing, and this paper explores how they can be used to protect quantum information as it is transformed by logical gates. Given a CSS code $\\mathcal{C}$, we derive conditions that are both necessary and sufficient for a transversal diagonal physical operator $U_Z$ to preserve $\\mathcal{C}$ and induce $U_L$. The group of $Z$-stabilizers in a CSS code $\\mathcal{C}$ is determined by the dual of a classical $[n, k_1]$ binary code $\\math"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2204.13176","kind":"arxiv","version":1},"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/2204.13176/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":"2204.13176","created_at":"2026-07-05T04:18:31.740361+00:00"},{"alias_kind":"arxiv_version","alias_value":"2204.13176v1","created_at":"2026-07-05T04:18:31.740361+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2204.13176","created_at":"2026-07-05T04:18:31.740361+00:00"},{"alias_kind":"pith_short_12","alias_value":"RTIAM7FNL6ZH","created_at":"2026-07-05T04:18:31.740361+00:00"},{"alias_kind":"pith_short_16","alias_value":"RTIAM7FNL6ZHHICW","created_at":"2026-07-05T04:18:31.740361+00:00"},{"alias_kind":"pith_short_8","alias_value":"RTIAM7FN","created_at":"2026-07-05T04:18:31.740361+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":0,"sample":[{"citing_arxiv_id":"2601.08568","citing_title":"Asymptotically good CSS codes that realize the logical transversal Clifford group fault-tolerantly","ref_index":12,"is_internal_anchor":false}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/RTIAM7FNL6ZHHICWMDPG2HANKC","json":"https://pith.science/pith/RTIAM7FNL6ZHHICWMDPG2HANKC.json","graph_json":"https://pith.science/api/pith-number/RTIAM7FNL6ZHHICWMDPG2HANKC/graph.json","events_json":"https://pith.science/api/pith-number/RTIAM7FNL6ZHHICWMDPG2HANKC/events.json","paper":"https://pith.science/paper/RTIAM7FN"},"agent_actions":{"view_html":"https://pith.science/pith/RTIAM7FNL6ZHHICWMDPG2HANKC","download_json":"https://pith.science/pith/RTIAM7FNL6ZHHICWMDPG2HANKC.json","view_paper":"https://pith.science/paper/RTIAM7FN","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2204.13176&json=true","fetch_graph":"https://pith.science/api/pith-number/RTIAM7FNL6ZHHICWMDPG2HANKC/graph.json","fetch_events":"https://pith.science/api/pith-number/RTIAM7FNL6ZHHICWMDPG2HANKC/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/RTIAM7FNL6ZHHICWMDPG2HANKC/action/timestamp_anchor","attest_storage":"https://pith.science/pith/RTIAM7FNL6ZHHICWMDPG2HANKC/action/storage_attestation","attest_author":"https://pith.science/pith/RTIAM7FNL6ZHHICWMDPG2HANKC/action/author_attestation","sign_citation":"https://pith.science/pith/RTIAM7FNL6ZHHICWMDPG2HANKC/action/citation_signature","submit_replication":"https://pith.science/pith/RTIAM7FNL6ZHHICWMDPG2HANKC/action/replication_record"}},"created_at":"2026-07-05T04:18:31.740361+00:00","updated_at":"2026-07-05T04:18:31.740361+00:00"}