# Effects of noise on the overparametrization of quantum neural networks

arXiv: https://arxiv.org/abs/2302.05059 · HTML: https://mlsanity.com/paper/2302.05059/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- Builds on 28 of its 88 references that are in the map.
- First posted on arXiv 10 Feb 2023.

## Builds on (most cited first, 28 of 28)

- [Reconciling modern machine learning practice and the bias-variance trade-off](https://mlsanity.com/paper/1812.11118/) (2018)
- [A Convergence Theory for Deep Learning via Over-Parameterization](https://mlsanity.com/paper/1811.03962/) (2018)
- [Variational Quantum Algorithms](https://mlsanity.com/paper/2012.09265/) (2020)
- [Gradient Descent Provably Optimizes Over-parameterized Neural Networks](https://mlsanity.com/paper/1810.02054/) (2018)
- [Barren plateaus in quantum neural network training landscapes](https://mlsanity.com/paper/1803.11173/) (2018)
- [Gradient Descent Finds Global Minima of Deep Neural Networks](https://mlsanity.com/paper/1811.03804/) (2018)
- [The power of quantum neural networks](https://mlsanity.com/paper/2011.00027/) (2020)
- [Parameterized quantum circuits as machine learning models](https://mlsanity.com/paper/1906.07682/) (2019)
- [Quantum Machine Learning](https://mlsanity.com/paper/1611.09347/) (2016)
- [High-dimensional dynamics of generalization error in neural networks](https://mlsanity.com/paper/1710.03667/) (2017)
- [Noise-Induced Barren Plateaus in Variational Quantum Algorithms](https://mlsanity.com/paper/2007.14384/) (2020)
- [Quantum advantage in learning from experiments](https://mlsanity.com/paper/2112.00778/) (2021)
- [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)
- [Theory of overparametrization in quantum neural networks](https://mlsanity.com/paper/2109.11676/) (2021)
- [Provably efficient machine learning for quantum many-body problems](https://mlsanity.com/paper/2106.12627/) (2021)
- [SGD Learns Over-parameterized Networks that Provably Generalize on Linearly Separable Data](https://mlsanity.com/paper/1710.10174/) (2017)
- [The jamming transition as a paradigm to understand the loss landscape of deep neural networks](https://mlsanity.com/paper/1809.09349/) (2018)
- [Theory for Equivariant Quantum Neural Networks](https://mlsanity.com/paper/2210.08566/) (2022)
- [Theoretical Guarantees for Permutation-Equivariant Quantum Neural Networks](https://mlsanity.com/paper/2210.09974/) (2022)
- [Representation Learning via Quantum Neural Tangent Kernels](https://mlsanity.com/paper/2111.04225/) (2021)
- [Capacity and quantum geometry of parametrized quantum circuits](https://mlsanity.com/paper/2102.01659/) (2021)
- [Analytic theory for the dynamics of wide quantum neural networks](https://mlsanity.com/paper/2203.16711/) (2022)
- [Symmetric Pruning in Quantum Neural Networks](https://mlsanity.com/paper/2208.14057/) (2022)
- [Learning Unitaries by Gradient Descent](https://mlsanity.com/paper/2001.11897/) (2020)
- [A Convergence Theory for Over-parameterized Variational Quantum Eigensolvers](https://mlsanity.com/paper/2205.12481/) (2022)
- [Qibo: a framework for quantum simulation with hardware acceleration](https://mlsanity.com/paper/2009.01845/) (2020)
- [Inference-Based Quantum Sensing](https://mlsanity.com/paper/2206.09919/) (2022)

## Built on by (most cited first, 9 of 9)

- [Barren Plateaus in Variational Quantum Computing](https://mlsanity.com/paper/2405.00781/) (2024)
- [Quantum Machine Learning on Near-Term Quantum Devices: Current State of Supervised and Unsupervised Techniques for Real-World Applications](https://mlsanity.com/paper/2307.00908/) (2023)
- [Generalization of Quantum Machine Learning Models Using Quantum Fisher Information Metric](https://mlsanity.com/paper/2303.13462/) (2023)
- [Forecasting steam mass flow in power plants using the parallel hybrid network](https://mlsanity.com/paper/2307.09483/) (2023)
- [On the universality of Sn-equivariant k-body gates](https://mlsanity.com/paper/2303.00728/) (2023)
- [Can Hessian-Based Insights Support Fault Diagnosis in Attention-based Models?](https://mlsanity.com/paper/2506.07871/) (2025)
- [Statistical Analysis of Quantum State Learning Process in Quantum Neural Networks](https://mlsanity.com/paper/2309.14980/) (2023)
- [Quantum Support Vector Regression for Robust Anomaly Detection](https://mlsanity.com/paper/2505.01012/) (2025)
- [Adaptive directional gradients for parameterised quantum circuits](https://mlsanity.com/paper/2606.09734/) (2026)

## Abstract (arXiv)

Overparametrization is one of the most surprising and notorious phenomena in machine learning. Recently, there have been several efforts to study if, and how, Quantum Neural Networks (QNNs) acting in the absence of hardware noise can be overparametrized. In particular, it has been proposed that a QNN can be defined as overparametrized if it has enough parameters to explore all available directions in state space. That is, if the rank of the Quantum Fisher Information Matrix (QFIM) for the QNN's output state is saturated. Here, we explore how the presence of noise affects the overparametrization phenomenon. Our results show that noise can "turn on" previously-zero eigenvalues of the QFIM. This enables the parametrized state to explore directions that were otherwise inaccessible, thus potentially turning an overparametrized QNN into an underparametrized one. For small noise levels, the QNN is quasi-overparametrized, as large eigenvalues coexists with small ones. Then, we prove that as the magnitude of noise increases all the eigenvalues of the QFIM become exponentially suppressed, indicating that the state becomes insensitive to any change in the parameters. As such, there is a pull-and-tug effect where noise can enable new directions, but also suppress the sensitivity to parameter updates. Finally, our results imply that current QNN capacity measures are ill-defined when hardware noise is present.