arXiv:2603.06466 · v4

Completeness for Prime-Dimensional Phase-Affine Circuits

首次提交 2026-03-07 00:51

版本更新 2026-11-04 16:05

雷达观测 2026-11-05 12:13

摘要

Equational reasoning about circuits underpins quantum-circuit optimisation and verification. In the qubit CNOT-dihedral fragment, phase polynomials and layered normal forms support 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 F_d 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 yields linear, quadratic, and cubic calculi. Their binomial-basis identities expose the mixed diagonal gates forced by affine transport. These calculi have unique phase-affine normal forms, and the resulting equational theories are complete: semantic equality coincides with derivable equality.

作者备注：In Proceedings QPL 2026, arXiv:2611.04069

期刊信息：EPTCS 456, 2026, pp. 40-127