{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:NUPG24RHA2EDICX4W6CPZJPHVD","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"83044fc23e214d2a82b4369a9b8c2076b198d18d586460bd1d3d5a46ec4bdf1f","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"quant-ph","submitted_at":"2015-01-12T16:41:46Z","title_canon_sha256":"7d4f5f34b63f2cd7f5c605b40dac2a7889304b4a67be7710cdc2bea165806e98"},"schema_version":"1.0","source":{"id":"1501.02708","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1501.02708","created_at":"2026-05-18T02:21:51Z"},{"alias_kind":"arxiv_version","alias_value":"1501.02708v2","created_at":"2026-05-18T02:21:51Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1501.02708","created_at":"2026-05-18T02:21:51Z"},{"alias_kind":"pith_short_12","alias_value":"NUPG24RHA2ED","created_at":"2026-05-18T12:29:34Z"},{"alias_kind":"pith_short_16","alias_value":"NUPG24RHA2EDICX4","created_at":"2026-05-18T12:29:34Z"},{"alias_kind":"pith_short_8","alias_value":"NUPG24RH","created_at":"2026-05-18T12:29:34Z"}],"graph_snapshots":[{"event_id":"sha256:72091a0bc931dcc94ab6625b2406b7c93d801410e592fc21f227ded441b2698d","target":"graph","created_at":"2026-05-18T02:21:51Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"We show that any unitary operator on the $d_A\\times d_B$ system ($d_A\\ge 2$) can be decomposed into the product of at most $4d_A-5$ controlled unitary operators. The number can be reduced to $2d_A-1$ when $d_A$ is a power of two. We also prove that three controlled unitaries can implement a bipartite complex permutation operator, and discuss the connection to an analogous result on classical reversible circuits. We further show that any $n$-partite unitary on the space $\\mathbb{C}^{d_1}\\otimes...\\otimes\\mathbb{C}^{d_n}$ is the product of at most $[2\\prod^{n-1}_{j=1}(2d_j-2)-1]$ controlled unit","authors_text":"Lin Chen, Li Yu","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"quant-ph","submitted_at":"2015-01-12T16:41:46Z","title":"Decomposition of bipartite and multipartite unitary gates into the product of controlled unitary gates"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1501.02708","kind":"arxiv","version":2},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:adb14f2978007515e6a24fcca46d78467598e39a6e5199d29272b9a73806397e","target":"record","created_at":"2026-05-18T02:21:51Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"83044fc23e214d2a82b4369a9b8c2076b198d18d586460bd1d3d5a46ec4bdf1f","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"quant-ph","submitted_at":"2015-01-12T16:41:46Z","title_canon_sha256":"7d4f5f34b63f2cd7f5c605b40dac2a7889304b4a67be7710cdc2bea165806e98"},"schema_version":"1.0","source":{"id":"1501.02708","kind":"arxiv","version":2}},"canonical_sha256":"6d1e6d72270688340afcb784fca5e7a8daed2478edd9c7925b4a01ab3bff3df0","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"6d1e6d72270688340afcb784fca5e7a8daed2478edd9c7925b4a01ab3bff3df0","first_computed_at":"2026-05-18T02:21:51.391404Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:21:51.391404Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"ZLIxlBiI5b5zbmXv8Cupl//P4hNQidyh/xwnzrS2zBaoovYl/4yHSqF7vEYX+h+3FuT92cLhub7mttB5lBDhAA==","signature_status":"signed_v1","signed_at":"2026-05-18T02:21:51.392090Z","signed_message":"canonical_sha256_bytes"},"source_id":"1501.02708","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:adb14f2978007515e6a24fcca46d78467598e39a6e5199d29272b9a73806397e","sha256:72091a0bc931dcc94ab6625b2406b7c93d801410e592fc21f227ded441b2698d"],"state_sha256":"888788713a898112c4c6feb1419e058c240c70c6db0a7f94dfb4e5cf9537c3c0"}