Simple qudit ZX and ZH calculi, via integrals

TL;DR

This paper introduces a unified ZXH-calculus framework for qudit quantum operations, utilizing integral-based semantic maps to facilitate analysis and rewrite rules for stabilizer and multicharacter fragments.

Contribution

It presents a novel integral-based semantic approach for qudit ZX and ZH calculi, enabling unified analysis of unitary circuits and measurements across any fixed dimension D>1.

Findings

Semantic maps for ZX and ZH diagrams using discrete measures

Rewrite rules for stabiliser fragment of ZX calculus

Rewrite rules for multicharacter fragment of ZH calculus

Abstract

The ZX calculus and ZH calculus use diagrams to denote and compute properties of quantum operations, using `rewrite rules' to transform between diagrams which denote the same operator through a functorial semantic map. Different semantic maps give rise to different rewrite systems, which may prove more convenient for different purposes. Using discrete measures, we describe semantic maps for ZX and ZH diagrams, well-suited to analyse unitary circuits and measurements on qudits of any fixed dimension D>1 as a single `ZXH-calculus'. We demonstrate rewrite rules for the `stabiliser fragment' of the ZX calculus and a `multicharacter fragment' of the ZH calculus.