{"paper":{"title":"Completeness for Prime-Dimensional Phase-Affine Circuits","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"quant-ph","authors_text":"Colin Blake","submitted_at":"2026-03-06T16:51:19Z","abstract_excerpt":"Equational reasoning about circuits underpins quantum-circuit optimisation and verification. The qubit CNOT-dihedral fragment achieves this through phase polynomials, layered normal forms, and a complete equational theory; we develop the corresponding theory for prime-dimensional qudits, where basis labels, value controls, and phase exponents share prime-field arithmetic. We first describe reversible affine circuits over Fd as transformations x->Ax+b, with an affine normal form extending Lafont's linear normal form by translations. Adjoining finite-angle diagonal phases by polynomial degree yi"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2603.06466","kind":"arxiv","version":2},"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/2603.06466/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"}