Paper
Completeness for Prime-Dimensional Phase-Affine Circuits
Authors
Colin Blake
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.
Metadata
Related papers
Cosmic Shear in Effective Field Theory at Two-Loop Order: Revisiting $S_8$ in Dark Energy Survey Data
Shi-Fan Chen, Joseph DeRose, Mikhail M. Ivanov, Oliver H. E. Philcox • 2026-03-30
Stop Probing, Start Coding: Why Linear Probes and Sparse Autoencoders Fail at Compositional Generalisation
Vitória Barin Pacela, Shruti Joshi, Isabela Camacho, Simon Lacoste-Julien, Da... • 2026-03-30
SNID-SAGE: A Modern Framework for Interactive Supernova Classification and Spectral Analysis
Fiorenzo Stoppa, Stephen J. Smartt • 2026-03-30
Acoustic-to-articulatory Inversion of the Complete Vocal Tract from RT-MRI with Various Audio Embeddings and Dataset Sizes
Sofiane Azzouz, Pierre-André Vuissoz, Yves Laprie • 2026-03-30
Rotating black hole shadows in metric-affine bumblebee gravity
Jose R. Nascimento, Ana R. M. Oliveira, Albert Yu. Petrov, Paulo J. Porfírio,... • 2026-03-30
Raw Data (Debug)
{
"raw_xml": "<entry>\n <id>http://arxiv.org/abs/2603.06466v1</id>\n <title>Completeness for Prime-Dimensional Phase-Affine Circuits</title>\n <updated>2026-03-06T16:51:19Z</updated>\n <link href='https://arxiv.org/abs/2603.06466v1' rel='alternate' type='text/html'/>\n <link href='https://arxiv.org/pdf/2603.06466v1' rel='related' title='pdf' type='application/pdf'/>\n <summary>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.</summary>\n <category scheme='http://arxiv.org/schemas/atom' term='quant-ph'/>\n <published>2026-03-06T16:51:19Z</published>\n <arxiv:primary_category term='quant-ph'/>\n <author>\n <name>Colin Blake</name>\n </author>\n </entry>"
}