# Equivalence of quantum barren plateaus to cost concentration and narrow gorges

arXiv: https://arxiv.org/abs/2104.05868 · HTML: https://mlsanity.com/paper/2104.05868/

## Key facts

- Cited by 53 papers in the map (data through 14 Oct 2026).
- Builds on 19 of its 68 references that are in the map.
- First posted on arXiv 12 Apr 2021.

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

- [Variational Quantum Algorithms](https://mlsanity.com/paper/2012.09265/) (2020)
- [Barren plateaus in quantum neural network training landscapes](https://mlsanity.com/paper/1803.11173/) (2018)
- [The power of quantum neural networks](https://mlsanity.com/paper/2011.00027/) (2020)
- [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)
- [Efficient Learning for Deep Quantum Neural Networks](https://mlsanity.com/paper/1902.10445/) (2019)
- [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)
- [Layerwise learning for quantum neural networks](https://mlsanity.com/paper/2006.14904/) (2020)
- [Effect of barren plateaus on gradient-free optimization](https://mlsanity.com/paper/2011.12245/) (2020)
- [Noisy intermediate-scale quantum (NISQ) algorithms](https://mlsanity.com/paper/2101.08448/) (2021)
- [TensorFlow Quantum: A Software Framework for Quantum Machine Learning](https://mlsanity.com/paper/2003.02989/) (2020)
- [Stochastic gradient descent for hybrid quantum-classical optimization](https://mlsanity.com/paper/1910.01155/) (2019)
- [Learning to learn with quantum neural networks via classical neural networks](https://mlsanity.com/paper/1907.05415/) (2019)
- [Trainability of Dissipative Perceptron-Based Quantum Neural Networks](https://mlsanity.com/paper/2005.12458/) (2020)
- [On barren plateaus and cost function locality in variational quantum algorithms](https://mlsanity.com/paper/2011.10530/) (2020)
- [Low-depth gradient measurements can improve convergence in variational hybrid quantum-classical algorithms](https://mlsanity.com/paper/1901.05374/) (2019)
- [Quantum Optimization for Training Quantum Neural Networks](https://mlsanity.com/paper/2103.17047/) (2021)

## Built on by (most cited first, 30 of 53)

- [Challenges and Opportunities in Quantum Machine Learning](https://mlsanity.com/paper/2303.09491/) (2023)
- [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)
- [Does provable absence of barren plateaus imply classical simulability?](https://mlsanity.com/paper/2312.09121/) (2023)
- [Theory for Equivariant Quantum Neural Networks](https://mlsanity.com/paper/2210.08566/) (2022)
- [Group-Invariant Quantum Machine Learning](https://mlsanity.com/paper/2205.02261/) (2022)
- [Theoretical Guarantees for Permutation-Equivariant Quantum Neural Networks](https://mlsanity.com/paper/2210.09974/) (2022)
- [Representation Theory for Geometric Quantum Machine Learning](https://mlsanity.com/paper/2210.07980/) (2022)
- [Recent Advances for Quantum Neural Networks in Generative Learning](https://mlsanity.com/paper/2206.03066/) (2022)
- [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)
- [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 the relation between trainability and dequantization of variational quantum learning models](https://mlsanity.com/paper/2406.07072/) (2024)
- [Problem-Dependent Power of Quantum Neural Networks on Multi-Class Classification](https://mlsanity.com/paper/2301.01597/) (2022)
- [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)
- [Optimal training of variational quantum algorithms without barren plateaus](https://mlsanity.com/paper/2104.14543/) (2021)
- [Variational quantum simulation: a case study for understanding warm starts](https://mlsanity.com/paper/2404.10044/) (2024)
- [Graph neural network initialisation of quantum approximate optimisation](https://mlsanity.com/paper/2111.03016/) (2021)
- [Adaptive shot allocation for fast convergence in variational quantum algorithms](https://mlsanity.com/paper/2108.10434/) (2021)
- [Quantum neural networks form Gaussian processes](https://mlsanity.com/paper/2305.09957/) (2023)
- [Quantum Boltzmann machine learning of ground-state energies](https://mlsanity.com/paper/2410.12935/) (2024)
- [Numerical evidence against advantage with quantum fidelity kernels on classical data](https://mlsanity.com/paper/2211.16551/) (2022)
- [Investigating and Mitigating Barren Plateaus in Variational Quantum Circuits: A Survey](https://mlsanity.com/paper/2407.17706/) (2024)
- [Reinforcement Learning for Variational Quantum Circuits Design](https://mlsanity.com/paper/2409.05475/) (2024)
- [Natural gradient and parameter estimation for quantum Boltzmann machines](https://mlsanity.com/paper/2410.24058/) (2024)
- [Expressibility-induced Concentration of Quantum Neural Tangent Kernels](https://mlsanity.com/paper/2311.04965/) (2023)

## Abstract (arXiv)

Optimizing parameterized quantum circuits (PQCs) is the leading approach to make use of near-term quantum computers. However, very little is known about the cost function landscape for PQCs, which hinders progress towards quantum-aware optimizers. In this work, we investigate the connection between three different landscape features that have been observed for PQCs: (1) exponentially vanishing gradients (called barren plateaus), (2) exponential cost concentration about the mean, and (3) the exponential narrowness of minina (called narrow gorges). We analytically prove that these three phenomena occur together, i.e., when one occurs then so do the other two. A key implication of this result is that one can numerically diagnose barren plateaus via cost differences rather than via the computationally more expensive gradients. More broadly, our work shows that quantum mechanics rules out certain cost landscapes (which otherwise would be mathematically possible), and hence our results are interesting from a quantum foundations perspective.