# Completeness of the ZX-Calculus

arXiv: https://arxiv.org/abs/1903.06035 · HTML: https://mlsanity.com/paper/1903.06035/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 13 Mar 2019.

## Built on by (most cited first, 2 of 2)

- [Analyzing the barren plateau phenomenon in training quantum neural networks with the ZX-calculus](https://mlsanity.com/paper/2102.01828/) (2021)
- [A supervised hybrid quantum machine learning solution to the emergency escape routing problem](https://mlsanity.com/paper/2307.15682/) (2023)

## Abstract (arXiv)

The ZX-Calculus is a graphical language for diagrammatic reasoning in quantum mechanics and quantum information theory. It comes equipped with an equational presentation. We focus here on a very important property of the language: completeness, which roughly ensures the equational theory captures all of quantum mechanics. We first improve on the known-to-be-complete presentation for the so-called Clifford fragment of the language - a restriction that is not universal - by adding some axioms. Thanks to a system of back-and-forth translation between the ZX-Calculus and a third-party complete graphical language, we prove that the provided axiomatisation is complete for the first approximately universal fragment of the language, namely Clifford+T. We then prove that the expressive power of this presentation, though aimed at achieving completeness for the aforementioned restriction, extends beyond Clifford+T, to a class of diagrams that we call linear with Clifford+T constants. We use another version of the third-party language - and an adapted system of back-and-forth translation - to complete the language for the ZX-Calculus as a whole, that is, with no restriction. We briefly discuss the added axioms, and finally, we provide a complete axiomatisation for an altered version of the language which involves an additional generator, making the presentation simpler.