{"paper":{"title":"A Compass on the Quantum State Sphere: The Hopf Ansatz for Arbitrary Pure-State Optimization","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"quant-ph","authors_text":"Guangxi Li, Ruge Lin","submitted_at":"2026-07-15T18:00:41Z","abstract_excerpt":"Optimizing arbitrary quantum state vectors is like navigating the unit sphere in Hilbert space: beyond target reachability, optimization asks for coordinates, local distance information, and measurable directions. We introduce the Hopf ansatz, a binary-tree circuit for arbitrary normalized real and complex state vectors. Internal angles steer probability between subtrees, leaf phases carry the complex degrees of freedom, and the same tree gives state preparation and an explicit inverse map from amplitudes to physical angles. Together these structures act as a compass for the search: the invers"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.14231","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/2607.14231/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"}