# Effect of barren plateaus on gradient-free optimization

arXiv: https://arxiv.org/abs/2011.12245 · HTML: https://mlsanity.com/paper/2011.12245/

## Key facts

- Cited by 80 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 24 Nov 2020.

## Built on by (most cited first, 30 of 80)

- [Generalization in quantum machine learning from few training data](https://mlsanity.com/paper/2111.05292/) (2021)
- [Challenges and Opportunities in Quantum Machine Learning](https://mlsanity.com/paper/2303.09491/) (2023)
- [Connecting ansatz expressibility to gradient magnitudes and barren plateaus](https://mlsanity.com/paper/2101.02138/) (2021)
- [Barren Plateaus in Variational Quantum Computing](https://mlsanity.com/paper/2405.00781/) (2024)
- [Theory of overparametrization in quantum neural networks](https://mlsanity.com/paper/2109.11676/) (2021)
- [Exponential concentration in quantum kernel methods](https://mlsanity.com/paper/2208.11060/) (2022)
- [Theory for Equivariant Quantum Neural Networks](https://mlsanity.com/paper/2210.08566/) (2022)
- [Equivalence of quantum barren plateaus to cost concentration and narrow gorges](https://mlsanity.com/paper/2104.05868/) (2021)
- [Understanding quantum machine learning also requires rethinking generalization](https://mlsanity.com/paper/2306.13461/) (2023)
- [Escaping from the Barren Plateau via Gaussian Initializations in Deep Variational Quantum Circuits](https://mlsanity.com/paper/2203.09376/) (2022)
- [Equivariant quantum circuits for learning on weighted graphs](https://mlsanity.com/paper/2205.06109/) (2022)
- [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)
- [A comprehensive review of Quantum Machine Learning: from NISQ to Fault Tolerance](https://mlsanity.com/paper/2401.11351/) (2024)
- [Entangled Datasets for Quantum Machine Learning](https://mlsanity.com/paper/2109.03400/) (2021)
- [On fundamental aspects of quantum extreme learning machines](https://mlsanity.com/paper/2312.15124/) (2023)
- [Enhancing Quantum Adversarial Robustness by Randomized Encodings](https://mlsanity.com/paper/2212.02531/) (2022)
- [A semi-agnostic ansatz with variable structure for quantum machine learning](https://mlsanity.com/paper/2103.06712/) (2021)
- [Dynamical simulation via quantum machine learning with provable generalization](https://mlsanity.com/paper/2204.10269/) (2022)
- [Can Error Mitigation Improve Trainability of Noisy Variational Quantum Algorithms?](https://mlsanity.com/paper/2109.01051/) (2021)
- [Avoiding Barren Plateaus with Classical Deep Neural Networks](https://mlsanity.com/paper/2205.13418/) (2022)
- [Reduction of finite sampling noise in quantum neural networks](https://mlsanity.com/paper/2306.01639/) (2023)
- [Adaptive shot allocation for fast convergence in variational quantum algorithms](https://mlsanity.com/paper/2108.10434/) (2021)
- [Fock State-enhanced Expressivity of Quantum Machine Learning Models](https://mlsanity.com/paper/2107.05224/) (2021)
- [Quantum Advantage Seeker with Kernels (QuASK): a software framework to speed up the research in quantum machine learning](https://mlsanity.com/paper/2206.15284/) (2022)
- [Laziness, Barren Plateau, and Noise in Machine Learning](https://mlsanity.com/paper/2206.09313/) (2022)
- [Investigating and Mitigating Barren Plateaus in Variational Quantum Circuits: A Survey](https://mlsanity.com/paper/2407.17706/) (2024)
- [Optimizing Quantum Variational Circuits with Deep Reinforcement Learning](https://mlsanity.com/paper/2109.03188/) (2021)
- [Reinforcement Learning for Variational Quantum Circuits Design](https://mlsanity.com/paper/2409.05475/) (2024)

## Abstract (arXiv)

Barren plateau landscapes correspond to gradients that vanish exponentially in the number of qubits. Such landscapes have been demonstrated for variational quantum algorithms and quantum neural networks with either deep circuits or global cost functions. For obvious reasons, it is expected that gradient-based optimizers will be significantly affected by barren plateaus. However, whether or not gradient-free optimizers are impacted is a topic of debate, with some arguing that gradient-free approaches are unaffected by barren plateaus. Here we show that, indeed, gradient-free optimizers do not solve the barren plateau problem. Our main result proves that cost function differences, which are the basis for making decisions in a gradient-free optimization, are exponentially suppressed in a barren plateau. Hence, without exponential precision, gradient-free optimizers will not make progress in the optimization. We numerically confirm this by training in a barren plateau with several gradient-free optimizers (Nelder-Mead, Powell, and COBYLA algorithms), and show that the numbers of shots required in the optimization grows exponentially with the number of qubits.