# Trained quantum neural networks are Gaussian processes

arXiv: https://arxiv.org/abs/2402.08726 · HTML: https://mlsanity.com/paper/2402.08726/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 28 references that are in the map.
- First posted on arXiv 13 Feb 2024.

## Builds on (most cited first, 11 of 11)

- [Deep Learning in Neural Networks: An Overview](https://mlsanity.com/paper/1404.7828/) (2014)
- [Wide Neural Networks of Any Depth Evolve as Linear Models Under Gradient Descent](https://mlsanity.com/paper/1902.06720/) (2019)
- [Variational Quantum Algorithms](https://mlsanity.com/paper/2012.09265/) (2020)
- [Supervised learning with quantum enhanced feature spaces](https://mlsanity.com/paper/1804.11326/) (2018)
- [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)
- [A rigorous and robust quantum speed-up in supervised machine learning](https://mlsanity.com/paper/2010.02174/) (2020)
- [Theory of overparametrization in quantum neural networks](https://mlsanity.com/paper/2109.11676/) (2021)
- [Six Lectures on Linearized Neural Networks](https://mlsanity.com/paper/2308.13431/) (2023)
- [Learning quantum data with the quantum Earth Mover's distance](https://mlsanity.com/paper/2101.03037/) (2021)
- [Quantum Lazy Training](https://mlsanity.com/paper/2202.08232/) (2022)

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

- [Provable and scalable quantum Gaussian processes for quantum learning](https://mlsanity.com/paper/2605.00099/) (2026)
- [Architectures and random properties of symplectic quantum circuits](https://mlsanity.com/paper/2405.10264/) (2024)
- [Quantum-Enhanced Neural Contextual Bandit Algorithms](https://mlsanity.com/paper/2601.02870/) (2026)
- [Quantitative Gaussian-Process limits of Tensor Programs](https://mlsanity.com/paper/2607.06290/) (2026)
- [Mean-field limit from general mixtures of experts to quantum neural networks](https://mlsanity.com/paper/2501.14660/) (2025)
- [Spectral Anatomy of Quantum Gaussian Process Kernels](https://mlsanity.com/paper/2605.30952/) (2026)

## Abstract (arXiv)

We study quantum neural networks made by parametric one-qubit gates and fixed two-qubit gates in the limit of infinite width, where the generated function is the expectation value of the sum of single-qubit observables over all the qubits. First, we prove that the probability distribution of the function generated by the untrained network with randomly initialized parameters converges in distribution to a Gaussian process whenever each measured qubit is correlated only with few other measured qubits. Then, we analytically characterize the training of the network via gradient descent with square loss on supervised learning problems. We prove that, as long as the network is not affected by barren plateaus, the trained network can perfectly fit the training set and that the probability distribution of the function generated after training still converges in distribution to a Gaussian process. Finally, we consider the statistical noise of the measurement at the output of the network and prove that a polynomial number of measurements is sufficient for all the previous results to hold and that the network can always be trained in polynomial time.