# A universal completion of the ZX-calculus

arXiv: https://arxiv.org/abs/1706.09877 · HTML: https://mlsanity.com/paper/1706.09877/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 29 Jun 2017.

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

- [Kindergarden quantum mechanics graduates (...or how I learned to stop gluing LEGO together and love the ZX-calculus)](https://mlsanity.com/paper/2102.10984/) (2021)
- [Optimizing Quantum Circuits via ZX Diagrams using Reinforcement Learning and Graph Neural Networks](https://mlsanity.com/paper/2504.03429/) (2025)

## Abstract (arXiv)

In this paper, we give a universal completion of the ZX-calculus for the whole of pure qubit quantum mechanics. This proof is based on the completeness of another graphical language: the ZW-calculus, with direct translations between these two graphical systems.