Graphical Algebraic Geometry: From Ideals and Varieties to Quantum Calculi

TL;DR

This paper introduces Graphical Algebraic Geometry (GAG), a diagrammatic language framework that unifies algebraic structures and quantum calculi, enabling intuitive reasoning and complexity analysis.

Contribution

It develops a universal, complete diagrammatic language for algebraic and quantum structures, connecting GAG with #CSP problems and the ZH calculus.

Findings

GAG can recast #CSP as rewrite problems, which are #P-hard.

GAG provides a complete rewrite system for polynomial networks.

Computing qudit ZH amplitudes requires only a constant number of GAG oracle queries.

Abstract

We introduce Graphical Algebraic Geometry (GAG), a family of diagrammatic languages extending the Graphical Linear Algebra programme. We construct several languages within this family and prove that they are universal and complete for the corresponding (co)span semantics of commutative algebras and affine varieties. This framework provides clear graphical representations of algebraic structures -- such as polynomials, ideals, and varieties -- enabling intuitive yet rigorous diagrammatic reasoning. We showcase two practical viewpoints on GAG. First, we show that instances of counting constraint satisfaction problem (#CSP) are recast as rewrite problems of closed diagrams in GAG. This means that deciding rewritability in GAG is #P-hard, and GAG can be viewed as a complete and compositional rewrite system for networks of polynomial constraints. Second, we characterize the qudit ZH calculus, a diagrammatic language for quantum computation, as an extension of Graphical Algebraic Geometry. This establishes the correspondence that Graphical Algebraic Geometry is to the ZH calculus what Graphical Linear Algebra is to the ZX calculus. Using this construction, we show that computing amplitudes in qudit ZH requires only a constant number of queries to a GAG oracle.