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Completeness for Prime-Dimensional Phase-Affine Circuits

Colin Blake

TL;DR

This work presents 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, and proves completeness: semantic equality coincides with derivable equality.

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.

Completeness for Prime-Dimensional Phase-Affine Circuits

TL;DR

This work presents 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, and proves completeness: semantic equality coincides with derivable equality.

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.
Paper Structure (26 sections, 92 theorems, 140 equations, 5 figures, 2 tables)

This paper contains 26 sections, 92 theorems, 140 equations, 5 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Affine circuits
  3. The semantic PROP of reversible affine maps
  4. The diagrammatic PROP Aff_d
  5. Affine layers and normal forms
  6. Finite-angle phase fragments
  7. Binomial polynomials over Fd
  8. Semantic PROPs of affine maps with phase polynomials
  9. Diagrammatic PROPs for finite-angle phases
  10. Interpretations and soundness
  11. Normal forms, uniqueness, and completeness
  12. Conclusion
  13. Acknowledgements
  14. Derivations
  15. Derivations for the Affine fragment
  16. ...and 11 more sections

Key Result

Lemma 1

Every axiom in $\textup{Aff}_d$ holds under $\left\llbracket \cdot \right\rrbracket_{Aff}$.

Figures (5)

  • Figure 1: The ruleset $\textup{Aff}_d$ for circuits of affine transformations over $\mathbb{F}_d$.
  • Figure 2: LinPhase$_d$ extends $\textup{Aff}_d$. It presents the PROP $\textbf{LinPhase}_d$.
  • Figure 3: $\textup{QuadPhase}_d$ extends $\textup{LinPhase}_d$. It presents the PROP $\textbf{QuadPhase}_d$.
  • Figure 4: $\textup{CubicPhase}_d$ extends $\textup{QuadPhase}_d$. It presents the PROP $\textbf{CubicPhase}_d$.
  • Figure 5: The ruleset lafont_boolean_circuits_2003 for circuits of linear transformations over $\mathrm{GL}(\mathrm{\mathbb{F}_d})$. We include \ref{['lafont-8']} because in Lafont's presentation the labels are in the field, which in $\mathbb{F}_d$ means taken modulo $d$.

Theorems & Definitions (206)

  • Definition 1
  • Definition 2
  • Definition 3
  • Remark 1
  • Definition 4
  • Definition 5
  • Definition 6
  • Definition 7
  • Lemma 1
  • Lemma 2
  • ...and 196 more