Completeness for Prime-Dimensional Phase-Affine Circuits
Abstract
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.
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BibTeX
@misc{blake2026completeness,
title = {{Completeness for Prime-Dimensional Phase-Affine Circuits}},
author = {Colin Blake},
howpublished = {HAL},
year = {2026},
month = jul,
eprint = {2603.06466},
archivePrefix = {arXiv},
url = {https://hal.science/hal-05540565v2},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Completeness for Prime-Dimensional Phase-Affine Circuits
AU - Colin Blake
PY - 2026
DA - 2026/07/06
PB - HAL
UR - https://hal.science/hal-05540565v2
AB - 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.
ER -