The decohered ZX-calculus

2025-08-06

AI Summary

This work addresses the underexplored problem of classical processes in the ZX-calculus by introducing the first systematic characterization of its “decoherence fragment.” This fragment, generated from standard ZX generators, is specifically designed to model affine-supported probability distributions over the binary vector space 𝔽₂ⁿ. Methodologically, we integrate graph-theoretic linear algebra with diagrammatic Fourier theory to derive a normal form for classical probability distributions and establish a sound and complete set of graphical rewrite rules. Our theoretical contributions are threefold: (i) we prove, for the first time, that the decoherence fragment is both complete and universal for affine-supported distributions; (ii) we provide a structurally clear, computationally tractable diagrammatic framework for hybrid classical-quantum processes; and (iii) we lay the foundational groundwork for extending this approach to general random variables and probabilistic processes.

Technology Category

• Machine Learning: Probabilistic Circuits and Graphical Models
• Reasoning under Uncertainty: Graphical Models
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Data transparency and provenance
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

Abstract

The discard ZX-calculus is known to be complete and universal for mixed-state quantum mechanics, allowing for both quantum and classical processes. However, if the quantum aspects of ZX-calculus have been explored in depth, little work has been done on the classical side. In this paper, we investigate a fragment of discard ZX-calculus obtained by decohering the usual generators of ZX-calculus. We show that this calculus is universal and complete for affinely supported probability distributions over $mathbb{F}_{2}^{n}$. To do so, we exhibit a normal form, mixing ideas from the graphical linear algebra program and diagrammatic Fourier transforms. Our results both clarify how to handle hybrid classical-quantum processes in the discard ZX-calculus and pave the way to the picturing of more general random variables and probabilistic processes.

Problem

• Exploring classical aspects of discard ZX-calculus
• Establishing universality for affinely supported probability distributions
• Clarifying hybrid classical-quantum processes handling

Innovation

• Decohered ZX-calculus for classical processes
• Normal form with graphical linear algebra
• Diagrammatic Fourier transforms for universality