Completeness for Prime-Dimensional Phase-Affine Circuits

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 yields linear, quadratic (odd prime), and cubic (prime greater than 3) calculi whose binomial-basis identities expose the mixed diagonal gates forced by affine transport. These calculi have unique phase-affine normal forms and are complete: semantic equality coincides with derivable equality, giving a prime-dimensional phase-polynomial analogue of the CNOT-dihedral equational theory.