Completeness for Prime-Dimensional Phase-Affine Circuits

TL;DR

This paper extends the complete equational theory of CNOT-dihedral circuits from qubits to prime-dimensional qudits, providing a normal form and completeness proof for affine and phase polynomial circuits.

Contribution

It introduces a compact PROP for prime-dimensional affine circuits, generalizes the CNOT-dihedral theory to qudits, and establishes a complete normal form with semantic and syntactic equality.

Findings

Established a normal form for prime-dimensional affine circuits.

Proved completeness: semantic equality equals derivable equality.

Derived uniform transport rules using binomial-basis identities.

Abstract

Equational reasoning about circuits is central in quantum software for validation, optimisation, and verification. For qubits, the CNOT-dihedral fragment supports efficient rewriting via phase polynomials and layered normal forms, yielding a complete and practically effective equational theory. In this work we generalise that CNOT-dihedral picture from qubits to prime-dimensional qudits. We present a compact PROP for reversible affine circuits over a prime field, with a strict symmetric monoidal semantics into the affine group and a Lafont-style affine normal form. We then adjoin finite-angle diagonal phase generators and organise them by polynomial degree, obtaining linear, quadratic (odd prime), and cubic (prime greater than 3) calculi. Using binomial-basis identities we derive uniform transport rules, establish unique phase-affine normal forms, and prove completeness: semantic equality coincides with derivable equality. This yields a prime-dimensional, phase-polynomial-aligned generalisation of the CNOT-dihedral equational theory.