# Theory of overparametrization in quantum neural networks

arXiv: https://arxiv.org/abs/2109.11676 · HTML: https://mlsanity.com/paper/2109.11676/

## Key facts

- Cited by 100 papers in the map (data through 14 Oct 2026).
- Builds on 39 of its 140 references that are in the map.
- First posted on arXiv 23 Sep 2021.

## Builds on (most cited first, 30 of 39)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [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)
- [Supervised learning with quantum enhanced feature spaces](https://mlsanity.com/paper/1804.11326/) (2018)
- [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)
- [A Survey of Deep Learning Techniques for Autonomous Driving](https://mlsanity.com/paper/1910.07738/) (2019)
- [The power of quantum neural networks](https://mlsanity.com/paper/2011.00027/) (2020)
- [Learning and Generalization in Overparameterized Neural Networks, Going Beyond Two Layers](https://mlsanity.com/paper/1811.04918/) (2018)
- [Power of data in quantum machine learning](https://mlsanity.com/paper/2011.01938/) (2020)
- [Parameterized quantum circuits as machine learning models](https://mlsanity.com/paper/1906.07682/) (2019)
- [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)
- [Towards Understanding the Role of Over-Parametrization in Generalization of Neural Networks](https://mlsanity.com/paper/1805.12076/) (2018)
- [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)
- [Information-theoretic bounds on quantum advantage in machine learning](https://mlsanity.com/paper/2101.02464/) (2021)
- [The Inductive Bias of Quantum Kernels](https://mlsanity.com/paper/2106.03747/) (2021)
- [Effect of barren plateaus on gradient-free optimization](https://mlsanity.com/paper/2011.12245/) (2020)
- [SGD Learns Over-parameterized Networks that Provably Generalize on Linearly Separable Data](https://mlsanity.com/paper/1710.10174/) (2017)
- [Noisy intermediate-scale quantum (NISQ) algorithms](https://mlsanity.com/paper/2101.08448/) (2021)
- [Trainability of Dissipative Perceptron-Based Quantum Neural Networks](https://mlsanity.com/paper/2005.12458/) (2020)
- [Equivalence of quantum barren plateaus to cost concentration and narrow gorges](https://mlsanity.com/paper/2104.05868/) (2021)
- [Complexity Theoretic Limitations on Learning Halfspaces](https://mlsanity.com/paper/1505.05800/) (2015)
- [Capacity and quantum geometry of parametrized quantum circuits](https://mlsanity.com/paper/2102.01659/) (2021)
- [Entangled Datasets for Quantum Machine Learning](https://mlsanity.com/paper/2109.03400/) (2021)
- [Complexity of Training ReLU Neural Network](https://mlsanity.com/paper/1809.10787/) (2018)

## Built on by (most cited first, 30 of 100)

- [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)
- [Barren Plateaus in Variational Quantum Computing](https://mlsanity.com/paper/2405.00781/) (2024)
- [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)
- [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)
- [Representation Theory for Geometric Quantum Machine Learning](https://mlsanity.com/paper/2210.07980/) (2022)
- [Subtleties in the trainability of quantum machine learning models](https://mlsanity.com/paper/2110.14753/) (2021)
- [Quantum Deep Hedging](https://mlsanity.com/paper/2303.16585/) (2023)
- [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)
- [Problem-Dependent Power of Quantum Neural Networks on Multi-Class Classification](https://mlsanity.com/paper/2301.01597/) (2022)
- [Hybrid quantum ResNet for car classification and its hyperparameter optimization](https://mlsanity.com/paper/2205.04878/) (2022)
- [Tensor networks for interpretable and efficient quantum-inspired machine learning](https://mlsanity.com/paper/2311.11258/) (2023)
- [A Convergence Theory for Over-parameterized Variational Quantum Eigensolvers](https://mlsanity.com/paper/2205.12481/) (2022)
- [Quantum tangent kernel](https://mlsanity.com/paper/2111.02951/) (2021)
- [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)
- [Variational quantum simulation: a case study for understanding warm starts](https://mlsanity.com/paper/2404.10044/) (2024)
- [Reduction of finite sampling noise in quantum neural networks](https://mlsanity.com/paper/2306.01639/) (2023)
- [Quantum neural networks form Gaussian processes](https://mlsanity.com/paper/2305.09957/) (2023)
- ['Eliv'agar: Efficient Quantum Circuit Search for Classification](https://mlsanity.com/paper/2401.09393/) (2024)
- [A Primer on Quantum Machine Learning](https://mlsanity.com/paper/2511.15969/) (2025)
- [Statistical Complexity of Quantum Learning](https://mlsanity.com/paper/2309.11617/) (2023)
- [Training-efficient density quantum machine learning](https://mlsanity.com/paper/2405.20237/) (2024)
- [Laziness, Barren Plateau, and Noise in Machine Learning](https://mlsanity.com/paper/2206.09313/) (2022)
- [Generalization of Quantum Machine Learning Models Using Quantum Fisher Information Metric](https://mlsanity.com/paper/2303.13462/) (2023)

## Abstract (arXiv)

The prospect of achieving quantum advantage with Quantum Neural Networks (QNNs) is exciting. Understanding how QNN properties (e.g., the number of parameters M) affect the loss landscape is crucial to the design of scalable QNN architectures. Here, we rigorously analyze the overparametrization phenomenon in QNNs with periodic structure. We define overparametrization as the regime where the QNN has more than a critical number of parameters Mc that allows it to explore all relevant directions in state space. Our main results show that the dimension of the Lie algebra obtained from the generators of the QNN is an upper bound for Mc, and for the maximal rank that the quantum Fisher information and Hessian matrices can reach. Underparametrized QNNs have spurious local minima in the loss landscape that start disappearing when M≥ Mc. Thus, the overparametrization onset corresponds to a computational phase transition where the QNN trainability is greatly improved by a more favorable landscape. We then connect the notion of overparametrization to the QNN capacity, so that when a QNN is overparametrized, its capacity achieves its maximum possible value. We run numerical simulations for eigensolver, compilation, and autoencoding applications to showcase the overparametrization computational phase transition. We note that our results also apply to variational quantum algorithms and quantum optimal control.