# Analyzing Convergence in Quantum Neural Networks: Deviations from Neural Tangent Kernels

arXiv: https://arxiv.org/abs/2303.14844 · HTML: https://mlsanity.com/paper/2303.14844/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Builds on 21 of its 39 references that are in the map.
- First posted on arXiv 26 Mar 2023.

## Builds on (most cited first, 21 of 21)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Neural Tangent Kernel: Convergence and Generalization in Neural Networks](https://mlsanity.com/paper/1806.07572/) (2018)
- [A Convergence Theory for Deep Learning via Over-Parameterization](https://mlsanity.com/paper/1811.03962/) (2018)
- [On Lazy Training in Differentiable Programming](https://mlsanity.com/paper/1812.07956/) (2018)
- [Barren plateaus in quantum neural network training landscapes](https://mlsanity.com/paper/1803.11173/) (2018)
- [On Exact Computation with an Infinitely Wide Neural Net](https://mlsanity.com/paper/1904.11955/) (2019)
- [Power of data in quantum machine learning](https://mlsanity.com/paper/2011.01938/) (2020)
- [On the Inductive Bias of Neural Tangent Kernels](https://mlsanity.com/paper/1905.12173/) (2019)
- [Towards moderate overparameterization: global convergence guarantees for training shallow neural networks](https://mlsanity.com/paper/1902.04674/) (2019)
- [Spectral Bias and Task-Model Alignment Explain Generalization in Kernel Regression and Infinitely Wide Neural Networks](https://mlsanity.com/paper/2006.13198/) (2020)
- [Theory of overparametrization in quantum neural networks](https://mlsanity.com/paper/2109.11676/) (2021)
- [On the Computational Efficiency of Training Neural Networks](https://mlsanity.com/paper/1410.1141/) (2014)
- [Representation Learning via Quantum Neural Tangent Kernels](https://mlsanity.com/paper/2111.04225/) (2021)
- [Representation Theory for Geometric Quantum Machine Learning](https://mlsanity.com/paper/2210.07980/) (2022)
- [Quantum Wasserstein Generative Adversarial Networks](https://mlsanity.com/paper/1911.00111/) (2019)
- [Enabling Multi-programming Mechanism for Quantum Computing in the NISQ Era](https://mlsanity.com/paper/2102.05321/) (2021)
- [Exponentially Many Local Minima in Quantum Neural Networks](https://mlsanity.com/paper/2110.02479/) (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)
- [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)

## Built on by (most cited first, 8 of 8)

- [Quantum-PEFT: Ultra parameter-efficient fine-tuning](https://mlsanity.com/paper/2503.05431/) (2025)
- [Double descent in quantum kernel methods](https://mlsanity.com/paper/2501.10077/) (2025)
- [Expressibility-induced Concentration of Quantum Neural Tangent Kernels](https://mlsanity.com/paper/2311.04965/) (2023)
- [Statistical Analysis of Quantum State Learning Process in Quantum Neural Networks](https://mlsanity.com/paper/2309.14980/) (2023)
- [New Insights on Unfolding and Fine-tuning Quantum Federated Learning](https://mlsanity.com/paper/2506.20016/) (2025)
- [PALQO: Physics-informed Model for Accelerating Large-scale Quantum Optimization](https://mlsanity.com/paper/2509.20733/) (2025)
- [Quantum-data-driven dynamical transition in quantum learning](https://mlsanity.com/paper/2410.01955/) (2024)
- [Zero-shot Quantum Neural Architecture Search](https://mlsanity.com/paper/2605.27410/) (2026)

## Abstract (arXiv)

A quantum neural network (QNN) is a parameterized mapping efficiently implementable on near-term Noisy Intermediate-Scale Quantum (NISQ) computers. It can be used for supervised learning when combined with classical gradient-based optimizers. Despite the existing empirical and theoretical investigations, the convergence of QNN training is not fully understood. Inspired by the success of the neural tangent kernels (NTKs) in probing into the dynamics of classical neural networks, a recent line of works proposes to study over-parameterized QNNs by examining a quantum version of tangent kernels. In this work, we study the dynamics of QNNs and show that contrary to popular belief it is qualitatively different from that of any kernel regression: due to the unitarity of quantum operations, there is a non-negligible deviation from the tangent kernel regression derived at the random initialization. As a result of the deviation, we prove the at-most sublinear convergence for QNNs with Pauli measurements, which is beyond the explanatory power of any kernel regression dynamics. We then present the actual dynamics of QNNs in the limit of over-parameterization. The new dynamics capture the change of convergence rate during training and implies that the range of measurements is crucial to the fast QNN convergence.