# ZX-calculus for the working quantum computer scientist

**Authors:** John van de Wetering  
**arXiv:** [2012.13966](https://arxiv.org/abs/2012.13966)  
**Date:** 2020-12-27  
**Summary page:** https://www.summarizepaper.com/en/arxiv-id/2012.13966v1/

## Key points

- Comprehensive review of the ZX-calculus, a graphical language for quantum computation
- Popular in quantum circuit optimization, surface codes, lattice surgery, measurement-based QC, and quantum foundations
- Gentle introduction to ZX-calculus for readers familiar with basic quantum computing concepts
- Equips readers to apply ZX-calculus for small computations on quantum circuits and states
- Overview of literature including discussions on Clifford computation and Gottesman-Knill theorem
- Extension allowing reasoning about Toffoli gates introduced
- Recent completeness theorems demonstrate capability to handle all reasoning using ZX-diagrams
- Exploration of categorical and algebraic origins of the language
- Extensions representing mixed states, measurement, classical control, and higher-dimensional qudits discussed

## Summary

The paper "ZX-calculus for the working quantum computer scientist" by John van de Wetering provides a comprehensive review of the ZX-calculus, a graphical language used for reasoning about quantum computation. This language has gained popularity in various areas such as quantum circuit optimization, surface codes, lattice surgery, measurement-based quantum computation, and quantum foundations. The first half of the review offers a gentle introduction to the ZX-calculus for readers familiar with basic concepts in quantum computing. The goal is to equip readers with enough knowledge to apply the ZX-calculus in their daily work for small computations on quantum circuits and states. The latter sections of the paper provide a condensed overview of literature on the subject. It includes discussions on Clifford computation and visually proving the Gottesman-Knill theorem. An extension of the ZX-calculus that allows for convenient reasoning about Toffoli gates is also introduced. Furthermore, recent completeness theorems for the ZX-calculus are covered, demonstrating its capability to handle all reasoning about quantum computation using ZX-diagrams. The paper also explores categorical and algebraic origins of this language and discusses several extensions that can represent mixed states, measurement, classical control, and higher-dimensional qudits. Overall,<DateTime>, this review serves as an invaluable resource for those looking to deepen their understanding and application of the ZX-calculus in various aspects of quantum computation.