# Optimal training of variational quantum algorithms without barren plateaus

arXiv: https://arxiv.org/abs/2104.14543 · HTML: https://mlsanity.com/paper/2104.14543/

## Key facts

- Cited by 16 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 64 references that are in the map.
- First posted on arXiv 29 Apr 2021.

## Builds on (most cited first, 14 of 14)

- [Variational Quantum Algorithms](https://mlsanity.com/paper/2012.09265/) (2020)
- [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)
- [Power of data in quantum machine learning](https://mlsanity.com/paper/2011.01938/) (2020)
- [The effect of data encoding on the expressive power of variational quantum machine learning models](https://mlsanity.com/paper/2008.08605/) (2020)
- [Supervised quantum machine learning models are kernel methods](https://mlsanity.com/paper/2101.11020/) (2021)
- [Noise-Induced Barren Plateaus in Variational Quantum Algorithms](https://mlsanity.com/paper/2007.14384/) (2020)
- [Quantum Natural Gradient](https://mlsanity.com/paper/1909.02108/) (2019)
- [Entanglement Induced Barren Plateaus](https://mlsanity.com/paper/2010.15968/) (2020)
- [Experimental Quantum Generative Adversarial Networks for Image Generation](https://mlsanity.com/paper/2010.06201/) (2020)
- [Noisy intermediate-scale quantum (NISQ) algorithms](https://mlsanity.com/paper/2101.08448/) (2021)
- [Equivalence of quantum barren plateaus to cost concentration and narrow gorges](https://mlsanity.com/paper/2104.05868/) (2021)
- [Capacity and quantum geometry of parametrized quantum circuits](https://mlsanity.com/paper/2102.01659/) (2021)
- [Long-time simulations with high fidelity on quantum hardware](https://mlsanity.com/paper/2102.04313/) (2021)

## Built on by (most cited first, 16 of 16)

- [Barren Plateaus in Variational Quantum Computing](https://mlsanity.com/paper/2405.00781/) (2024)
- [Does provable absence of barren plateaus imply classical simulability?](https://mlsanity.com/paper/2312.09121/) (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)
- [Quantum machine learning of large datasets using randomized measurements](https://mlsanity.com/paper/2108.01039/) (2021)
- [Variational quantum simulation: a case study for understanding warm starts](https://mlsanity.com/paper/2404.10044/) (2024)
- [Investigating and Mitigating Barren Plateaus in Variational Quantum Circuits: A Survey](https://mlsanity.com/paper/2407.17706/) (2024)
- [Stochastic Gradient Line Bayesian Optimization for Efficient Noise-Robust Optimization of Parameterized Quantum Circuits](https://mlsanity.com/paper/2111.07952/) (2021)
- [Quantum Natural Policy Gradients: Towards Sample-Efficient Reinforcement Learning](https://mlsanity.com/paper/2304.13571/) (2023)
- [LAWS: Look Around and Warm-Start Natural Gradient Descent for Quantum Neural Networks](https://mlsanity.com/paper/2205.02666/) (2022)
- [On the similarity of bandwidth-tuned quantum kernels and classical kernels](https://mlsanity.com/paper/2503.05602/) (2025)
- [Enhancing variational quantum algorithms by balancing training on classical and quantum hardware](https://mlsanity.com/paper/2503.16361/) (2025)
- [Enhancing the Trainability of Variational Quantum Circuits with Regularization Strategies](https://mlsanity.com/paper/2405.01606/) (2024)
- [Optimisation-free Classification and Density Estimation with Quantum Circuits](https://mlsanity.com/paper/2203.14452/) (2022)
- [On Quantum Natural Policy Gradients](https://mlsanity.com/paper/2401.08307/) (2024)
- [What a Reporting Convention Hides: A Matched-Budget Audit of Quantum Natural Gradient with an Exactly Computed Metric](https://mlsanity.com/paper/2610.09425/) (2026)

## Abstract (arXiv)

Variational quantum algorithms (VQAs) promise efficient use of near-term quantum computers. However, training VQAs often requires an extensive amount of time and suffers from the barren plateau problem where the magnitude of the gradients vanishes with increasing number of qubits. Here, we show how to optimally train VQAs for learning quantum states. Parameterized quantum circuits can form Gaussian kernels, which we use to derive adaptive learning rates for gradient ascent. We introduce the generalized quantum natural gradient that features stability and optimized movement in parameter space. Both methods together outperform other optimization routines in training VQAs. Our methods also excel at numerically optimizing driving protocols for quantum control problems. The gradients of the VQA do not vanish when the fidelity between the initial state and the state to be learned is bounded from below. We identify a VQA for quantum simulation with such a constraint that thus can be trained free of barren plateaus. Finally, we propose the application of Gaussian kernels for quantum machine learning.