{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:FOYX2XT5FGPDMANUNI5H5MSE2L","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":"5d3602654986fd15428180dd7c04980198824e56a6f57b6bf0322083db40b4ec","cross_cats_sorted":["cs.IT","math.IT","math.PR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"quant-ph","submitted_at":"2015-10-09T18:51:01Z","title_canon_sha256":"0fd8c0dd18b8996330945ca17edde24281f9246af7d486b68495f39e40c48dcf"},"schema_version":"1.0","source":{"id":"1510.02767","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1510.02767","created_at":"2026-05-18T01:30:41Z"},{"alias_kind":"arxiv_version","alias_value":"1510.02767v1","created_at":"2026-05-18T01:30:41Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1510.02767","created_at":"2026-05-18T01:30:41Z"},{"alias_kind":"pith_short_12","alias_value":"FOYX2XT5FGPD","created_at":"2026-05-18T12:29:19Z"},{"alias_kind":"pith_short_16","alias_value":"FOYX2XT5FGPDMANU","created_at":"2026-05-18T12:29:19Z"},{"alias_kind":"pith_short_8","alias_value":"FOYX2XT5","created_at":"2026-05-18T12:29:19Z"}],"graph_snapshots":[{"event_id":"sha256:6d074d6705f434531cc1673801b4e4724504a5144f85d6e8a7a7ec74405ce4d0","target":"graph","created_at":"2026-05-18T01:30:41Z","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":"A complex projective $t$-design is a configuration of vectors which is ``evenly distributed'' on a sphere in the sense that sampling uniformly from it reproduces the moments of Haar measure up to order $2t$. We show that the set of all $n$-qubit stabilizer states forms a complex projective $3$-design in dimension $2^n$. Stabilizer states had previously only been known to constitute $2$-designs. The main technical ingredient is a general recursion formula for the so-called frame potential of stabilizer states. To establish it, we need to compute the number of stabilizer states with pre-describe","authors_text":"David Gross, Richard Kueng","cross_cats":["cs.IT","math.IT","math.PR"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"quant-ph","submitted_at":"2015-10-09T18:51:01Z","title":"Qubit stabilizer states are complex projective 3-designs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1510.02767","kind":"arxiv","version":1},"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:b0ebba77fc7388ec2fcc5c07a01100e4dcf9c70a9ea3387fc2e73ad4c662022b","target":"record","created_at":"2026-05-18T01:30:41Z","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":"5d3602654986fd15428180dd7c04980198824e56a6f57b6bf0322083db40b4ec","cross_cats_sorted":["cs.IT","math.IT","math.PR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"quant-ph","submitted_at":"2015-10-09T18:51:01Z","title_canon_sha256":"0fd8c0dd18b8996330945ca17edde24281f9246af7d486b68495f39e40c48dcf"},"schema_version":"1.0","source":{"id":"1510.02767","kind":"arxiv","version":1}},"canonical_sha256":"2bb17d5e7d299e3601b46a3a7eb244d2d78bd3fed05e7711378ddabe7db29992","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"2bb17d5e7d299e3601b46a3a7eb244d2d78bd3fed05e7711378ddabe7db29992","first_computed_at":"2026-05-18T01:30:41.535974Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:30:41.535974Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"ji0m19o2NUAbDoqF5kmfmg7ieee3vU9RDZjm/66a22zTcEEDKAADunQjwuX8n0S3ITS5J56gqPgu9LGL6vRKBQ==","signature_status":"signed_v1","signed_at":"2026-05-18T01:30:41.536658Z","signed_message":"canonical_sha256_bytes"},"source_id":"1510.02767","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:b0ebba77fc7388ec2fcc5c07a01100e4dcf9c70a9ea3387fc2e73ad4c662022b","sha256:6d074d6705f434531cc1673801b4e4724504a5144f85d6e8a7a7ec74405ce4d0"],"state_sha256":"73966392ff694919fbc9700ac02f30678a3e4484fe980c3516d179ee3dd48ae4"}