Completeness for Prime-Dimensional Phase-Affine Circuits

• URL: http://arxiv.org/abs/2603.06466v1
• Date: Fri, 06 Mar 2026 16:51:19 GMT
• Title: Completeness for Prime-Dimensional Phase-Affine Circuits
• Abstract summary: We present a compact PROP for CNOT-dihedral picture from qubits to prime-dimensional qudits.<n>We derive uniform transport rules, establish unique phase-affine normal forms, and prove: semantic equality coincides with derivable equality.<n>This yields a prime-dimensional, phase-polynomial-aligned generalisation of the CNOT-dihedral equational theory.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

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