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Enriching Diagrams with Algebraic Operations

Alejandro Villoria, Henning Basold, Alfons Laarman

TL;DR

This work develops a framework for enriching diagrammatic reasoning in monoidal categories with algebraic operations by leveraging Eilenberg–Moore algebras for a monad, focusing on affine monads. It proves a free enrichment construction that preserves symmetric monoidal structure and lifts an adjunction to enriched monoidal categories, enabling a semantic bridge to $ extbf{Alg}^T$-enriched ZX-diagrams. The Distribution monad $ extbf{D}$ is used to richly encode probabilistic mixtures, yielding enriched ZX-diagrams whose interpretations are convex combinations of quantum maps, and providing rules to reason about noise in quantum systems. The approach yields a uniform, semantics-driven method to extend ZX-calculus to probabilistic and noisy quantum contexts, with soundness established and completeness left as an open area for future work. Potential impacts include automated reasoning about quantum noise, efficient diagrammatic representations of mixed processes, and extensions to other monads capturing diverse computational effects.

Abstract

In this paper, we extend diagrammatic reasoning in monoidal categories with algebraic operations and equations. We achieve this by considering monoidal categories that are enriched in the category of Eilenberg-Moore algebras for a monad. Under the condition that this monad is monoidal and affine, we construct an adjunction between symmetric monoidal categories and symmetric monoidal categories enriched over algebras for the monad. This allows us to devise an extension, and its semantics, of the ZX-calculus with probabilistic choices by freely enriching over convex algebras, which are the algebras of the finite distribution monad. We show how this construction can be used for diagrammatic reasoning of noise in quantum systems.

Enriching Diagrams with Algebraic Operations

TL;DR

This work develops a framework for enriching diagrammatic reasoning in monoidal categories with algebraic operations by leveraging Eilenberg–Moore algebras for a monad, focusing on affine monads. It proves a free enrichment construction that preserves symmetric monoidal structure and lifts an adjunction to enriched monoidal categories, enabling a semantic bridge to -enriched ZX-diagrams. The Distribution monad is used to richly encode probabilistic mixtures, yielding enriched ZX-diagrams whose interpretations are convex combinations of quantum maps, and providing rules to reason about noise in quantum systems. The approach yields a uniform, semantics-driven method to extend ZX-calculus to probabilistic and noisy quantum contexts, with soundness established and completeness left as an open area for future work. Potential impacts include automated reasoning about quantum noise, efficient diagrammatic representations of mixed processes, and extensions to other monads capturing diverse computational effects.

Abstract

In this paper, we extend diagrammatic reasoning in monoidal categories with algebraic operations and equations. We achieve this by considering monoidal categories that are enriched in the category of Eilenberg-Moore algebras for a monad. Under the condition that this monad is monoidal and affine, we construct an adjunction between symmetric monoidal categories and symmetric monoidal categories enriched over algebras for the monad. This allows us to devise an extension, and its semantics, of the ZX-calculus with probabilistic choices by freely enriching over convex algebras, which are the algebras of the finite distribution monad. We show how this construction can be used for diagrammatic reasoning of noise in quantum systems.
Paper Structure (19 sections, 6 theorems, 27 equations, 6 figures)

This paper contains 19 sections, 6 theorems, 27 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Contributions
  3. Related Work
  4. Outline
  5. Background
  6. Algebraic Enrichment
  7. Enriched Monoidal Categories
  8. Applications: ZX-calculus
  9. Quantum Computing
  10. The ZX-calculus
  11. Enriching the Categories $\mathbf{Qubit}$ and $\mathbf{CPM(\mathbf{Qubit})}$
  12. Enriched ZX-diagrams and Their Interpretation
  13. Additional Rules for Enriched ZX-diagrams
  14. Discussion and Future Work
  15. Proof of \ref{['thm:pseudo-monoid-adjunction']}
  16. ...and 4 more sections

Key Result

lemma thmcounterlemma

Let $(\mathbf{V},\otimes,I)$ be a closed symmetric monoidal category with coproducts. Then the hom-functor $\mathbf{V} (I,-) \colon \mathbf{V} \to \mathbf{Set}$ has a left adjoint $F$ that sends a set $X$ to $F(X)=\coprod_X I$, the $X$-th fold copower of $I$. Moreover, $F$ is a strong morphism of sy

Figures (6)

  • Figure 1: Left: Probabilistic mix of a gate $G$ with an error $E$. Right: Interaction of tensor and convex sum, where double wires visually indicate a tensor product
  • Figure 2: ZX-diagrams generators and how to compose them.
  • Figure 3: Standard interpretation of ZX-diagrams.
  • Figure 4: ZX-calculus ruleset. All rules also hold when swapping the colors of the spiders. In $(eu)$ we omit the calculation of the angles, which can be found in vilmartOptimal.
  • Figure 5: Left: Diagrammatic representation of the depolarizing channel. Right: Diagrammatic representation of a mixture of two-qubit gates.
  • ...and 1 more figures

Theorems & Definitions (9)

  • lemma thmcounterlemma: borceux1994
  • theorem thmcountertheorem
  • proof
  • corollary thmcountercorollary
  • proof
  • theorem thmcountertheorem
  • proof
  • theorem thmcountertheorem
  • corollary thmcountercorollary