mlsanityData to 7 Oct 2026

Connecting ansatz expressibility to gradient magnitudes and barren plateaus

arXiv 2101.02138 ↗First posted 6 Jan 2021quant-ph cs.LG stat.MLcited by 180

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Abstract (arXiv)

Parameterized quantum circuits serve as ans"atze for solving variational problems and provide a flexible paradigm for programming near-term quantum computers. Ideally, such ans"atze should be highly expressive so that a close approximation of the desired solution can be accessed. On the other hand, the ansatz must also have sufficiently large gradients to allow for training. Here, we derive a fundamental relationship between these two essential properties: expressibility and trainability. This is done by extending the well established barren plateau phenomenon, which holds for ans"atze that form exact 2-designs, to arbitrary ans"atze. Specifically, we calculate the variance in the cost gradient in terms of the expressibility of the ansatz, as measured by its distance from being a 2-design. Our resulting bounds indicate that highly expressive ans"atze exhibit flatter cost landscapes and therefore will be harder to train. Furthermore, we provide numerics illustrating the effect of expressiblity on gradient scalings, and we discuss the implications for designing strategies to avoid barren plateaus.

Key facts. Cited by 180 papers in the map (data through 7 Oct 2026). Builds on 15 of its 71 references that are in the map. First posted on arXiv 6 Jan 2021.

Builds on

15 of 71 references.

References · 15

  1. 2020Variational Quantum Algorithmscited by 687
  2. 2018Barren plateaus in quantum neural network training landscapescited by 624
  3. 2020The power of quantum neural networkscited by 377
  4. 2016Quantum Machine Learningcited by 325
  5. 2020Noise-Induced Barren Plateaus in Variational Quantum Algorithmscited by 192
  6. 2020Absence of Barren Plateaus in Quantum Convolutional Neural Networkscited by 124
  7. 2020Entanglement Induced Barren Plateauscited by 101
  8. 2020Layerwise learning for quantum neural networkscited by 87
  9. 2020Effect of barren plateaus on gradient-free optimizationcited by 79
  10. 2020TensorFlow Quantum: A Software Framework for Quantum Machine Learningcited by 72
  11. 2020Trainability of Dissipative Perceptron-Based Quantum Neural Networkscited by 56
  12. 2019Learning to learn with quantum neural networks via classical neural networkscited by 55
3 more
  1. 2020On barren plateaus and cost function locality in variational quantum algorithmscited by 36
  2. 2020Reformulation of the No-Free-Lunch Theorem for Entangled Data Setscited by 28
  3. 2020Cost Function Dependent Barren Plateaus in Shallow Parametrized Quantum Circuitscited by 26

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2021: 20; 2022: 23; 2023: 29; 2024: 42; 2025: 33; 2026: 33.

180 papers.

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  1. 2021Generalization in quantum machine learning from few training datacited by 214
  2. 2023Challenges and Opportunities in Quantum Machine Learningcited by 202
  3. 2024Barren Plateaus in Variational Quantum Computingcited by 130
  4. 2021Theory of overparametrization in quantum neural networkscited by 100
  5. 2022Exponential concentration in quantum kernel methodscited by 89
  6. 2023Does provable absence of barren plateaus imply classical simulability?cited by 78
  7. 2022Theory for Equivariant Quantum Neural Networkscited by 60
  8. 2022Group-Invariant Quantum Machine Learningcited by 56
  9. 2021Equivalence of quantum barren plateaus to cost concentration and narrow gorgescited by 53
  10. 2022Theoretical Guarantees for Permutation-Equivariant Quantum Neural Networkscited by 52
  11. 2023Understanding quantum machine learning also requires rethinking generalizationcited by 48
  12. 2022Escaping from the Barren Plateau via Gaussian Initializations in Deep Variational Quantum Circuitscited by 42
38 more
  1. 2022Representation Theory for Geometric Quantum Machine Learningcited by 40
  2. 2021Recent advances for quantum classifierscited by 35
  3. 2023Trainability barriers and opportunities in quantum generative modelingcited by 35
  4. 2021Subtleties in the trainability of quantum machine learning modelscited by 33
  5. 2024Quantum Convolutional Neural Networks are Effectively Classically Simulablecited by 33
  6. 2022Bandwidth Enables Generalization in Quantum Kernel Modelscited by 32
  7. 2024A comprehensive review of Quantum Machine Learning: from NISQ to Fault Tolerancecited by 32
  8. 2021Analyzing the barren plateau phenomenon in training quantum neural networks with the ZX-calculuscited by 30
  9. 2023Machine Learning for Anomaly Detection in Particle Physicscited by 29
  10. 2021Entangled Datasets for Quantum Machine Learningcited by 26
  11. 2022Hybrid Quantum-Classical Generative Adversarial Network for High Resolution Image Generationcited by 22
  12. 2022Analytic theory for the dynamics of wide quantum neural networkscited by 21
  13. 2021The dilemma of quantum neural networkscited by 20
  14. 2022Problem-Dependent Power of Quantum Neural Networks on Multi-Class Classificationcited by 20
  15. 2024Learning shallow quantum circuitscited by 20
  16. 2021Variational inference with a quantum computercited by 18
  17. 2023On fundamental aspects of quantum extreme learning machinescited by 18
  18. 2023Explaining Quantum Circuits with Shapley Values: Towards Explainable Quantum Machine Learningcited by 17
  19. 2021A semi-agnostic ansatz with variable structure for quantum machine learningcited by 15
  20. 2022Dynamical simulation via quantum machine learning with provable generalizationcited by 15
  21. 2021Can Error Mitigation Improve Trainability of Noisy Variational Quantum Algorithms?cited by 14
  22. 2023An Empirical Comparison of Optimizers for Quantum Machine Learning with SPSA-based Gradientscited by 14
  23. 2024Variational quantum simulation: a case study for understanding warm startscited by 14
  24. 2022Avoiding Barren Plateaus with Classical Deep Neural Networkscited by 13
  25. 2021Adaptive shot allocation for fast convergence in variational quantum algorithmscited by 12
  26. 2023Reduction of finite sampling noise in quantum neural networkscited by 12
  27. 2024VQC-Based Reinforcement Learning with Data Re-uploading: Performance and Trainabilitycited by 12
  28. 2021Fock State-enhanced Expressivity of Quantum Machine Learning Modelscited by 11
  29. 2022Quantum Advantage Seeker with Kernels (QuASK): a software framework to speed up the research in quantum machine learningcited by 11
  30. 2024Quantum Boltzmann machine learning of ground-state energiescited by 11
  31. 2023Splitting and Parallelizing of Quantum Convolutional Neural Networks for Learning Translationally Symmetric Datacited by 10
  32. 2023Approximately Equivariant Quantum Neural Network for p4m Group Symmetries in Imagescited by 10
  33. 2024Training-efficient density quantum machine learningcited by 10
  34. 2023Effects of noise on the overparametrization of quantum neural networkscited by 9
  35. 2024Investigating and Mitigating Barren Plateaus in Variational Quantum Circuits: A Surveycited by 9
  36. 2024Natural gradient and parameter estimation for quantum Boltzmann machinescited by 9
  37. 2025QCPINN: Quantum-Classical Physics-Informed Neural Networks for Solving PDEscited by 9
  38. 2021Long-time simulations with high fidelity on quantum hardwarecited by 8

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