# Analyzing the barren plateau phenomenon in training quantum neural networks with the ZX-calculus

arXiv: https://arxiv.org/abs/2102.01828 · HTML: https://mlsanity.com/paper/2102.01828/

## Key facts

- Cited by 30 papers in the map (data through 14 Oct 2026).
- Builds on 19 of its 56 references that are in the map.
- First posted on arXiv 3 Feb 2021.

## Builds on (most cited first, 19 of 19)

- [Supervised learning with quantum enhanced feature spaces](https://mlsanity.com/paper/1804.11326/) (2018)
- [Barren plateaus in quantum neural network training landscapes](https://mlsanity.com/paper/1803.11173/) (2018)
- [Parameterized quantum circuits as machine learning models](https://mlsanity.com/paper/1906.07682/) (2019)
- [Noise-Induced Barren Plateaus in Variational Quantum Algorithms](https://mlsanity.com/paper/2007.14384/) (2020)
- [Connecting ansatz expressibility to gradient magnitudes and barren plateaus](https://mlsanity.com/paper/2101.02138/) (2021)
- [Absence of Barren Plateaus in Quantum Convolutional Neural Networks](https://mlsanity.com/paper/2011.02966/) (2020)
- [Quantum Natural Gradient](https://mlsanity.com/paper/1909.02108/) (2019)
- [Entanglement Induced Barren Plateaus](https://mlsanity.com/paper/2010.15968/) (2020)
- [Differentiable Learning of Quantum Circuit Born Machine](https://mlsanity.com/paper/1804.04168/) (2018)
- [Unsupervised Generative Modeling Using Matrix Product States](https://mlsanity.com/paper/1709.01662/) (2017)
- [Trainability of Dissipative Perceptron-Based Quantum Neural Networks](https://mlsanity.com/paper/2005.12458/) (2020)
- [Foundations for Near-Term Quantum Natural Language Processing](https://mlsanity.com/paper/2012.03755/) (2020)
- [Cost Function Dependent Barren Plateaus in Shallow Parametrized Quantum Circuits](https://mlsanity.com/paper/2001.00550/) (2020)
- [Graph-theoretic Simplification of Quantum Circuits with the ZX-calculus](https://mlsanity.com/paper/1902.03178/) (2019)
- [Phase Gadget Synthesis for Shallow Circuits](https://mlsanity.com/paper/1906.01734/) (2019)
- [QDNN: DNN with Quantum Neural Network Layers](https://mlsanity.com/paper/1912.12660/) (2019)
- [Diagrammatic Design and Study of Ans"atze for Quantum Machine Learning](https://mlsanity.com/paper/2011.11073/) (2020)
- [The ZX calculus is a language for surface code lattice surgery](https://mlsanity.com/paper/1704.08670/) (2017)
- [Completeness of the ZX-Calculus](https://mlsanity.com/paper/1903.06035/) (2019)

## Built on by (most cited first, 30 of 30)

- [Barren Plateaus in Variational Quantum Computing](https://mlsanity.com/paper/2405.00781/) (2024)
- [Exponential concentration in quantum kernel methods](https://mlsanity.com/paper/2208.11060/) (2022)
- [Does provable absence of barren plateaus imply classical simulability?](https://mlsanity.com/paper/2312.09121/) (2023)
- [Quantum Convolutional Neural Networks are Effectively Classically Simulable](https://mlsanity.com/paper/2408.12739/) (2024)
- [Trainability barriers and opportunities in quantum generative modeling](https://mlsanity.com/paper/2305.02881/) (2023)
- [Recent advances for quantum classifiers](https://mlsanity.com/paper/2108.13421/) (2021)
- [Subtleties in the trainability of quantum machine learning models](https://mlsanity.com/paper/2110.14753/) (2021)
- [Analytic theory for the dynamics of wide quantum neural networks](https://mlsanity.com/paper/2203.16711/) (2022)
- [Hybrid Quantum Classical Graph Neural Networks for Particle Track Reconstruction](https://mlsanity.com/paper/2109.12636/) (2021)
- [Variational inference with a quantum computer](https://mlsanity.com/paper/2103.06720/) (2021)
- [Benchmarking simulated and physical quantum processing units using quantum and hybrid algorithms](https://mlsanity.com/paper/2211.15631/) (2022)
- [An exponentially-growing family of universal quantum circuits](https://mlsanity.com/paper/2212.00736/) (2022)
- [Enhancing Quantum Adversarial Robustness by Randomized Encodings](https://mlsanity.com/paper/2212.02531/) (2022)
- [Tensor networks for quantum machine learning](https://mlsanity.com/paper/2303.11735/) (2023)
- [A semi-agnostic ansatz with variable structure for quantum machine learning](https://mlsanity.com/paper/2103.06712/) (2021)
- [Investigating and Mitigating Barren Plateaus in Variational Quantum Circuits: A Survey](https://mlsanity.com/paper/2407.17706/) (2024)
- [Expressibility-induced Concentration of Quantum Neural Tangent Kernels](https://mlsanity.com/paper/2311.04965/) (2023)
- [Generalization Error Bound for Quantum Machine Learning in NISQ Era -- A Survey](https://mlsanity.com/paper/2409.07626/) (2024)
- [Diagrammatic Differentiation for Quantum Machine Learning](https://mlsanity.com/paper/2103.07960/) (2021)
- [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)
- [Quantum Capsule Networks](https://mlsanity.com/paper/2201.01778/) (2022)
- [Quantum Extremal Learning](https://mlsanity.com/paper/2205.02807/) (2022)
- [Superposed parameterised quantum circuits](https://mlsanity.com/paper/2506.08749/) (2025)
- [Differentiating and Integrating ZX Diagrams with Applications to Quantum Machine Learning](https://mlsanity.com/paper/2201.13250/) (2022)
- [QSpeech: Low-Qubit Quantum Speech Application Toolkit](https://mlsanity.com/paper/2205.13221/) (2022)
- [Data-Dependent Generalization Bounds for Parameterized Quantum Models Under Noise](https://mlsanity.com/paper/2412.11451/) (2024)
- [Taming Barren Plateaus in Arbitrary Parameterized Quantum Circuits without Sacrificing Expressibility](https://mlsanity.com/paper/2511.13408/) (2025)
- [Decomposing Gradient Suppression in Barren Plateaus: Activity, Sign Organization, and Coupling](https://mlsanity.com/paper/2605.01319/) (2026)
- [Application of Quantum Tensor Networks for Protein Classification](https://mlsanity.com/paper/2403.06890/) (2024)
- [From Trainability Diagnostics to Optimization Claims: Boundaries and Controls in Variational Quantum Optimization](https://mlsanity.com/paper/2609.21243/) (2026)

## Abstract (arXiv)

In this paper, we propose a general scheme to analyze the gradient vanishing phenomenon, also known as the barren plateau phenomenon, in training quantum neural networks with the ZX-calculus. More precisely, we extend the barren plateaus theorem from unitary 2-design circuits to any parameterized quantum circuits under certain reasonable assumptions. The main technical contribution of this paper is representing certain integrations as ZX-diagrams and computing them with the ZX-calculus. The method is used to analyze four concrete quantum neural networks with different structures. It is shown that, for the hardware efficient ansatz and the MPS-inspired ansatz, there exist barren plateaus, while for the QCNN ansatz and the tree tensor network ansatz, there exists no barren plateau.