{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:JUVVSPCB6BKAAUBHFV4ZE6EZWI","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":"ef17c210285b5c84ff963100b457b80c57b9b8e0e47be12852d30534c039a087","cross_cats_sorted":["cs.DS","quant-ph"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CC","submitted_at":"2012-05-21T04:00:32Z","title_canon_sha256":"e9760e90a9a4e4a0ebd416cf07d733aac2e24dc60ece04c6d7b3be774c4d0ece"},"schema_version":"1.0","source":{"id":"1205.4484","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1205.4484","created_at":"2026-05-18T02:35:52Z"},{"alias_kind":"arxiv_version","alias_value":"1205.4484v3","created_at":"2026-05-18T02:35:52Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1205.4484","created_at":"2026-05-18T02:35:52Z"},{"alias_kind":"pith_short_12","alias_value":"JUVVSPCB6BKA","created_at":"2026-05-18T12:27:11Z"},{"alias_kind":"pith_short_16","alias_value":"JUVVSPCB6BKAAUBH","created_at":"2026-05-18T12:27:11Z"},{"alias_kind":"pith_short_8","alias_value":"JUVVSPCB","created_at":"2026-05-18T12:27:11Z"}],"graph_snapshots":[{"event_id":"sha256:1e24d04118fecb4fcc5ddc032cb8f446120524cb41259a98ae2c87e3a2c0a346","target":"graph","created_at":"2026-05-18T02:35:52Z","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 study the computational complexity of approximating the 2->q norm of linear operators (defined as ||A||_{2->q} = sup_v ||Av||_q/||v||_2), as well as connections between this question and issues arising in quantum information theory and the study of Khot's Unique Games Conjecture (UGC). We show the following:\n  1. For any constant even integer q>=4, a graph $G$ is a \"small-set expander\" if and only if the projector into the span of the top eigenvectors of G's adjacency matrix has bounded 2->q norm. As a corollary, a good approximation to the 2->q norm will refute the Small-Set Expansion Conj","authors_text":"Aram W. Harrow, Boaz Barak, David Steurer, Fernando G.S.L. Brand\\~ao, Jonathan A. Kelner, Yuan Zhou","cross_cats":["cs.DS","quant-ph"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CC","submitted_at":"2012-05-21T04:00:32Z","title":"Hypercontractivity, Sum-of-Squares Proofs, and their Applications"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1205.4484","kind":"arxiv","version":3},"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:e0ecfe8adbfa0ab67301c17ca832c66a11aec5934255bd0e81fed5f53ed4ddc2","target":"record","created_at":"2026-05-18T02:35:52Z","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":"ef17c210285b5c84ff963100b457b80c57b9b8e0e47be12852d30534c039a087","cross_cats_sorted":["cs.DS","quant-ph"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CC","submitted_at":"2012-05-21T04:00:32Z","title_canon_sha256":"e9760e90a9a4e4a0ebd416cf07d733aac2e24dc60ece04c6d7b3be774c4d0ece"},"schema_version":"1.0","source":{"id":"1205.4484","kind":"arxiv","version":3}},"canonical_sha256":"4d2b593c41f0540050272d79927899b2288ae7f028d84271b28fb9ee3ac2da44","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"4d2b593c41f0540050272d79927899b2288ae7f028d84271b28fb9ee3ac2da44","first_computed_at":"2026-05-18T02:35:52.763207Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:35:52.763207Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"KnQhZMiae0yOARkerja6YXNpOWci0kCIZL4sD+ERheudrjSJHWEj2K8y6cBL/Ypr4P47Xc1XW5kECWagYsVKBA==","signature_status":"signed_v1","signed_at":"2026-05-18T02:35:52.763852Z","signed_message":"canonical_sha256_bytes"},"source_id":"1205.4484","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:e0ecfe8adbfa0ab67301c17ca832c66a11aec5934255bd0e81fed5f53ed4ddc2","sha256:1e24d04118fecb4fcc5ddc032cb8f446120524cb41259a98ae2c87e3a2c0a346"],"state_sha256":"d4e3a08ceb1e7b1988d03bd8e5fdab8c5ef3b96daf0441d9928338cf1d3d713c"}