# The quantum cost function concentration dependency on the parametrization expressivity

arXiv: https://arxiv.org/abs/2301.06883 · HTML: https://mlsanity.com/paper/2301.06883/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 20 of its 49 references that are in the map.
- First posted on arXiv 17 Jan 2023.

## Builds on (most cited first, 20 of 20)

- [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 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)
- [Connecting ansatz expressibility to gradient magnitudes and barren plateaus](https://mlsanity.com/paper/2101.02138/) (2021)
- [Entanglement Induced Barren Plateaus](https://mlsanity.com/paper/2010.15968/) (2020)
- [Robust data encodings for quantum classifiers](https://mlsanity.com/paper/2003.01695/) (2020)
- [Layerwise learning for quantum neural networks](https://mlsanity.com/paper/2006.14904/) (2020)
- [Experimental Quantum Generative Adversarial Networks for Image Generation](https://mlsanity.com/paper/2010.06201/) (2020)
- [Effect of barren plateaus on gradient-free optimization](https://mlsanity.com/paper/2011.12245/) (2020)
- [Quantum Architecture Search via Deep Reinforcement Learning](https://mlsanity.com/paper/2104.07715/) (2021)
- [Learning to learn with quantum neural networks via classical neural networks](https://mlsanity.com/paper/1907.05415/) (2019)
- [On barren plateaus and cost function locality in variational quantum algorithms](https://mlsanity.com/paper/2011.10530/) (2020)
- [Hybrid quantum convolutional neural networks model for COVID-19 prediction using chest X-Ray images](https://mlsanity.com/paper/2102.06535/) (2021)
- [Natural Evolutionary Strategies for Variational Quantum Computation](https://mlsanity.com/paper/2012.00101/) (2020)
- [A hybrid quantum-classical neural network with deep residual learning](https://mlsanity.com/paper/2012.07772/) (2020)
- [Avoiding Barren Plateaus with Classical Deep Neural Networks](https://mlsanity.com/paper/2205.13418/) (2022)
- [Evolution strategies: Application in hybrid quantum-classical neural networks](https://mlsanity.com/paper/2205.08059/) (2022)
- [Restricting to the chip architecture maintains the quantum neural network accuracy](https://mlsanity.com/paper/2212.14426/) (2022)

## Built on by (most cited first, 6 of 6)

- [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)
- [Graph Neural Networks on Quantum Computers](https://mlsanity.com/paper/2405.17060/) (2024)
- [Quantum neural network with ensemble learning to mitigate barren plateaus and cost function concentration](https://mlsanity.com/paper/2402.06026/) (2024)
- [Barren plateaus are amplified by the dimension of qudits](https://mlsanity.com/paper/2405.08190/) (2024)
- [On the Design of Expressive and Trainable Pulse-based Quantum Machine Learning Models](https://mlsanity.com/paper/2508.05559/) (2025)

## Abstract (arXiv)

Although we are currently in the era of noisy intermediate scale quantum devices, several studies are being conducted with the aim of bringing machine learning to the quantum domain. Currently, quantum variational circuits are one of the main strategies used to build such models. However, despite its widespread use, we still do not know what are the minimum resources needed to create a quantum machine learning model. In this article, we analyze how the expressiveness of the parametrization affects the cost function. We analytically show that the more expressive the parametrization is, the more the cost function will tend to concentrate around a value that depends both on the chosen observable and on the number of qubits used. For this, we initially obtain a relationship between the expressiveness of the parametrization and the mean value of the cost function. Afterwards, we relate the expressivity of the parametrization with the variance of the cost function. Finally, we show some numerical simulation results that confirm our theoretical-analytical predictions. To the best of our knowledge, this is the first time that these two important aspects of quantum neural networks are explicitly connected.