mlsanityData to 7 Oct 2026

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

arXiv 2102.01828 ↗First posted 3 Feb 2021quant-ph cs.LGcited by 30

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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.

Key facts. Cited by 30 papers in the map (data through 7 Oct 2026). Builds on 19 of its 56 references that are in the map. First posted on arXiv 3 Feb 2021.

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19 of 56 references.

References · 19

  1. 2018Supervised learning with quantum enhanced feature spacescited by 654
  2. 2018Barren plateaus in quantum neural network training landscapescited by 624
  3. 2019Parameterized quantum circuits as machine learning modelscited by 330
  4. 2020Noise-Induced Barren Plateaus in Variational Quantum Algorithmscited by 192
  5. 2021Connecting ansatz expressibility to gradient magnitudes and barren plateauscited by 180
  6. 2020Absence of Barren Plateaus in Quantum Convolutional Neural Networkscited by 124
  7. 2019Quantum Natural Gradientcited by 104
  8. 2020Entanglement Induced Barren Plateauscited by 101
  9. 2018Differentiable Learning of Quantum Circuit Born Machinecited by 90
  10. 2017Unsupervised Generative Modeling Using Matrix Product Statescited by 87
  11. 2020Trainability of Dissipative Perceptron-Based Quantum Neural Networkscited by 56
  12. 2020Foundations for Near-Term Quantum Natural Language Processingcited by 51
7 more
  1. 2020Cost Function Dependent Barren Plateaus in Shallow Parametrized Quantum Circuitscited by 26
  2. 2019Graph-theoretic Simplification of Quantum Circuits with the ZX-calculuscited by 20
  3. 2019Phase Gadget Synthesis for Shallow Circuitscited by 8
  4. 2019QDNN: DNN with Quantum Neural Network Layerscited by 6
  5. 2020Diagrammatic Design and Study of Ans"atze for Quantum Machine Learningcited by 5
  6. 2017The ZX calculus is a language for surface code lattice surgerycited by 4
  7. 2019Completeness of the ZX-Calculuscited by 2

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2021: 7; 2022: 9; 2023: 4; 2024: 6; 2025: 2; 2026: 2.

30 papers.

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  1. 2024Barren Plateaus in Variational Quantum Computingcited by 130
  2. 2022Exponential concentration in quantum kernel methodscited by 89
  3. 2023Does provable absence of barren plateaus imply classical simulability?cited by 78
  4. 2021Recent advances for quantum classifierscited by 35
  5. 2023Trainability barriers and opportunities in quantum generative modelingcited by 35
  6. 2021Subtleties in the trainability of quantum machine learning modelscited by 33
  7. 2024Quantum Convolutional Neural Networks are Effectively Classically Simulablecited by 33
  8. 2022Analytic theory for the dynamics of wide quantum neural networkscited by 21
  9. 2021Hybrid Quantum Classical Graph Neural Networks for Particle Track Reconstructioncited by 20
  10. 2021Variational inference with a quantum computercited by 18
  11. 2022Benchmarking simulated and physical quantum processing units using quantum and hybrid algorithmscited by 18
  12. 2022An exponentially-growing family of universal quantum circuitscited by 17
18 more
  1. 2022Enhancing Quantum Adversarial Robustness by Randomized Encodingscited by 16
  2. 2023Tensor networks for quantum machine learningcited by 16
  3. 2021A semi-agnostic ansatz with variable structure for quantum machine learningcited by 15
  4. 2024Investigating and Mitigating Barren Plateaus in Variational Quantum Circuits: A Surveycited by 9
  5. 2023Expressibility-induced Concentration of Quantum Neural Tangent Kernelscited by 7
  6. 2024Generalization Error Bound for Quantum Machine Learning in NISQ Era -- A Surveycited by 7
  7. 2021Diagrammatic Differentiation for Quantum Machine Learningcited by 5
  8. 2021Kindergarden quantum mechanics graduates (...or how I learned to stop gluing LEGO together and love the ZX-calculus)cited by 4
  9. 2022Quantum Capsule Networkscited by 3
  10. 2022Quantum Extremal Learningcited by 3
  11. 2025Superposed parameterised quantum circuitscited by 2
  12. 2022Differentiating and Integrating ZX Diagrams with Applications to Quantum Machine Learningcited by 1
  13. 2022QSpeech: Low-Qubit Quantum Speech Application Toolkitcited by 1
  14. 2024Data-Dependent Generalization Bounds for Parameterized Quantum Models Under Noisecited by 1
  15. 2026Decomposing Gradient Suppression in Barren Plateaus: Activity, Sign Organization, and Couplingcited by 1
  16. 2024Application of Quantum Tensor Networks for Protein Classificationcited by 0
  17. 2025Taming Barren Plateaus in Arbitrary Parameterized Quantum Circuits without Sacrificing Expressibilitycited by 0
  18. 2026From Trainability Diagnostics to Optimization Claims: Boundaries and Controls in Variational Quantum Optimizationcited by 0