{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2005:CZYVBX3OB4NH424W2LHTCN7W4T","short_pith_number":"pith:CZYVBX3O","schema_version":"1.0","canonical_sha256":"167150df6e0f1a7e6b96d2cf3137f6e4da438695185c031c956c2c6928e094fc","source":{"kind":"arxiv","id":"quant-ph/0502070","version":1},"attestation_state":"computed","paper":{"title":"A geometric approach to quantum circuit lower bounds","license":"","headline":"","cross_cats":[],"primary_cat":"quant-ph","authors_text":"Michael A. Nielsen","submitted_at":"2005-02-11T23:54:58Z","abstract_excerpt":"What is the minimal size quantum circuit required to exactly implement a specified n-qubit unitary operation, U, without the use of ancilla qubits? We show that a lower bound on the minimal size is provided by the length of the minimal geodesic between U and the identity, I, where length is defined by a suitable Finsler metric on SU(2^n). The geodesic curves of such a metric have the striking property that once an initial position and velocity are set, the remainder of the geodesic is completely determined by a second order differential equation known as the geodesic equation. This is in contr"},"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/0502070","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"quant-ph","submitted_at":"2005-02-11T23:54:58Z","cross_cats_sorted":[],"title_canon_sha256":"80d381b200a45a840a75fa5b34ba78cda8ab8dfff20bdfa3ab00d5648438aac0","abstract_canon_sha256":"f3e33746e171ad0bba96333e7db7615313cdc5cf46b8a092b4c6e9a5aa23b594"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-04T14:46:55.313524Z","signature_b64":"RX73K3/tqNWp8hzhgzN9ioHmhwxyoVhykjlyyidfm/MEafW/TIU5QK+VuYg7rJIEKKgLCpkInl0SKmruPRa5DA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"167150df6e0f1a7e6b96d2cf3137f6e4da438695185c031c956c2c6928e094fc","last_reissued_at":"2026-07-04T14:46:55.313154Z","signature_status":"signed_v1","first_computed_at":"2026-07-04T14:46:55.313154Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A geometric approach to quantum circuit lower bounds","license":"","headline":"","cross_cats":[],"primary_cat":"quant-ph","authors_text":"Michael A. Nielsen","submitted_at":"2005-02-11T23:54:58Z","abstract_excerpt":"What is the minimal size quantum circuit required to exactly implement a specified n-qubit unitary operation, U, without the use of ancilla qubits? We show that a lower bound on the minimal size is provided by the length of the minimal geodesic between U and the identity, I, where length is defined by a suitable Finsler metric on SU(2^n). The geodesic curves of such a metric have the striking property that once an initial position and velocity are set, the remainder of the geodesic is completely determined by a second order differential equation known as the geodesic equation. This is in contr"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"quant-ph/0502070","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/quant-ph/0502070/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/0502070","created_at":"2026-07-04T14:46:55.313211+00:00"},{"alias_kind":"arxiv_version","alias_value":"quant-ph/0502070v1","created_at":"2026-07-04T14:46:55.313211+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.quant-ph/0502070","created_at":"2026-07-04T14:46:55.313211+00:00"},{"alias_kind":"pith_short_12","alias_value":"CZYVBX3OB4NH","created_at":"2026-07-04T14:46:55.313211+00:00"},{"alias_kind":"pith_short_16","alias_value":"CZYVBX3OB4NH424W","created_at":"2026-07-04T14:46:55.313211+00:00"},{"alias_kind":"pith_short_8","alias_value":"CZYVBX3O","created_at":"2026-07-04T14:46:55.313211+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":14,"internal_anchor_count":11,"sample":[{"citing_arxiv_id":"2606.23785","citing_title":"Controlled Chaos in 4D SCFTs","ref_index":81,"is_internal_anchor":true},{"citing_arxiv_id":"2606.03049","citing_title":"Holographic complexity of de-Sitter black holes","ref_index":55,"is_internal_anchor":true},{"citing_arxiv_id":"2606.02817","citing_title":"Nielsen complexity with multiple cost factors","ref_index":18,"is_internal_anchor":true},{"citing_arxiv_id":"2405.09628","citing_title":"Quantum Dynamics in Krylov Space: Methods and Applications","ref_index":145,"is_internal_anchor":true},{"citing_arxiv_id":"2311.18804","citing_title":"Holographic complexity of the Klebanov-Strassler background","ref_index":2,"is_internal_anchor":true},{"citing_arxiv_id":"2407.21278","citing_title":"Universal Euler-Cartan Circuits for Quantum Field Theories","ref_index":48,"is_internal_anchor":true},{"citing_arxiv_id":"2412.08925","citing_title":"Generalized CV Conjecture and Krylov Complexity in Two-Mode Hermitian Systems via Information Geometry","ref_index":7,"is_internal_anchor":true},{"citing_arxiv_id":"2505.11553","citing_title":"Holographic entanglement entropy and complexity for the cosmological braneworld model","ref_index":49,"is_internal_anchor":true},{"citing_arxiv_id":"2605.16507","citing_title":"Krylov complexity from a simple quantum mechanical model for a radiating black hole","ref_index":2,"is_internal_anchor":true},{"citing_arxiv_id":"2508.04486","citing_title":"Quantum circuit complexity and unsupervised machine learning of topological order","ref_index":68,"is_internal_anchor":true},{"citing_arxiv_id":"2509.14810","citing_title":"Krylov Complexity for Open Quantum System: Dissipation and Decoherence","ref_index":17,"is_internal_anchor":true},{"citing_arxiv_id":"2604.27858","citing_title":"Geometric complexity in thermodynamics","ref_index":32,"is_internal_anchor":false},{"citing_arxiv_id":"2604.23031","citing_title":"How fast can a quantum gate be? Exact speed limits from geometry","ref_index":28,"is_internal_anchor":false},{"citing_arxiv_id":"2604.14275","citing_title":"Generalized Complexity Distances and Non-Invertible Symmetries","ref_index":8,"is_internal_anchor":false}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/CZYVBX3OB4NH424W2LHTCN7W4T","json":"https://pith.science/pith/CZYVBX3OB4NH424W2LHTCN7W4T.json","graph_json":"https://pith.science/api/pith-number/CZYVBX3OB4NH424W2LHTCN7W4T/graph.json","events_json":"https://pith.science/api/pith-number/CZYVBX3OB4NH424W2LHTCN7W4T/events.json","paper":"https://pith.science/paper/CZYVBX3O"},"agent_actions":{"view_html":"https://pith.science/pith/CZYVBX3OB4NH424W2LHTCN7W4T","download_json":"https://pith.science/pith/CZYVBX3OB4NH424W2LHTCN7W4T.json","view_paper":"https://pith.science/paper/CZYVBX3O","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=quant-ph/0502070&json=true","fetch_graph":"https://pith.science/api/pith-number/CZYVBX3OB4NH424W2LHTCN7W4T/graph.json","fetch_events":"https://pith.science/api/pith-number/CZYVBX3OB4NH424W2LHTCN7W4T/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/CZYVBX3OB4NH424W2LHTCN7W4T/action/timestamp_anchor","attest_storage":"https://pith.science/pith/CZYVBX3OB4NH424W2LHTCN7W4T/action/storage_attestation","attest_author":"https://pith.science/pith/CZYVBX3OB4NH424W2LHTCN7W4T/action/author_attestation","sign_citation":"https://pith.science/pith/CZYVBX3OB4NH424W2LHTCN7W4T/action/citation_signature","submit_replication":"https://pith.science/pith/CZYVBX3OB4NH424W2LHTCN7W4T/action/replication_record"}},"created_at":"2026-07-04T14:46:55.313211+00:00","updated_at":"2026-07-04T14:46:55.313211+00:00"}