# Differentiating and Integrating ZX Diagrams with Applications to Quantum Machine Learning

arXiv: https://arxiv.org/abs/2201.13250 · HTML: https://mlsanity.com/paper/2201.13250/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 56 references that are in the map.
- First posted on arXiv 31 Jan 2022.

## Builds on (most cited first, 13 of 13)

- [Barren plateaus in quantum neural network training landscapes](https://mlsanity.com/paper/1803.11173/) (2018)
- [The Loss Surfaces of Multilayer Networks](https://mlsanity.com/paper/1412.0233/) (2014)
- [QNLP in Practice: Running Compositional Models of Meaning on a Quantum Computer](https://mlsanity.com/paper/2102.12846/) (2021)
- [Foundations for Near-Term Quantum Natural Language Processing](https://mlsanity.com/paper/2012.03755/) (2020)
- [lambeq: An Efficient High-Level Python Library for Quantum NLP](https://mlsanity.com/paper/2110.04236/) (2021)
- [Analyzing the barren plateau phenomenon in training quantum neural networks with the ZX-calculus](https://mlsanity.com/paper/2102.01828/) (2021)
- [Graph-theoretic Simplification of Quantum Circuits with the ZX-calculus](https://mlsanity.com/paper/1902.03178/) (2019)
- [Diagrammatic Design and Study of Ans"atze for Quantum Machine Learning](https://mlsanity.com/paper/2011.11073/) (2020)
- [Diagrammatic Differentiation for Quantum Machine Learning](https://mlsanity.com/paper/2103.07960/) (2021)
- [A Diagrammatic Axiomatisation for Qubit Entanglement](https://mlsanity.com/paper/1501.07082/) (2015)
- [The ZX calculus is a language for surface code lattice surgery](https://mlsanity.com/paper/1704.08670/) (2017)
- [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)
- [Generalised Mermin-type non-locality arguments](https://mlsanity.com/paper/1702.01772/) (2017)

## Built on by (most cited first, 1 of 1)

- [Hybrid quantum recurrent neural network for remaining useful life prediction of turbofan engines](https://mlsanity.com/paper/2504.20823/) (2025)

## Abstract (arXiv)

ZX-calculus has proved to be a useful tool for quantum technology with a wide range of successful applications. Most of these applications are of an algebraic nature. However, other tasks that involve differentiation and integration remain unreachable with current ZX techniques. Here we elevate ZX to an analytical perspective by realising differentiation and integration entirely within the framework of ZX-calculus. We explicitly illustrate the new analytic framework of ZX-calculus by applying it in context of quantum machine learning for the analysis of barren plateaus.