# Statistical Analysis of Quantum State Learning Process in Quantum Neural Networks

arXiv: https://arxiv.org/abs/2309.14980 · HTML: https://mlsanity.com/paper/2309.14980/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 29 of its 79 references that are in the map.
- First posted on arXiv 26 Sep 2023.

## Builds on (most cited first, 29 of 29)

- [Barren plateaus in quantum neural network training landscapes](https://mlsanity.com/paper/1803.11173/) (2018)
- [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)
- [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)
- [Variational Quantum Circuits for Deep Reinforcement Learning](https://mlsanity.com/paper/1907.00397/) (2019)
- [Theory of overparametrization in quantum neural networks](https://mlsanity.com/paper/2109.11676/) (2021)
- [Quantum machine learning beyond kernel methods](https://mlsanity.com/paper/2110.13162/) (2021)
- [Provably efficient machine learning for quantum many-body problems](https://mlsanity.com/paper/2106.12627/) (2021)
- [Layerwise learning for quantum neural networks](https://mlsanity.com/paper/2006.14904/) (2020)
- [Escaping from the Barren Plateau via Gaussian Initializations in Deep Variational Quantum Circuits](https://mlsanity.com/paper/2203.09376/) (2022)
- [Representation Learning via Quantum Neural Tangent Kernels](https://mlsanity.com/paper/2111.04225/) (2021)
- [Recent Advances for Quantum Neural Networks in Generative Learning](https://mlsanity.com/paper/2206.03066/) (2022)
- [Recent advances for quantum classifiers](https://mlsanity.com/paper/2108.13421/) (2021)
- [Theoretical Error Performance Analysis for Variational Quantum Circuit Based Functional Regression](https://mlsanity.com/paper/2206.04804/) (2022)
- [Exponentially Many Local Minima in Quantum Neural Networks](https://mlsanity.com/paper/2110.02479/) (2021)
- [Symmetric Pruning in Quantum Neural Networks](https://mlsanity.com/paper/2208.14057/) (2022)
- [The dilemma of quantum neural networks](https://mlsanity.com/paper/2106.04975/) (2021)
- [Sublinear quantum algorithms for training linear and kernel-based classifiers](https://mlsanity.com/paper/1904.02276/) (2019)
- [Power and limitations of single-qubit native quantum neural networks](https://mlsanity.com/paper/2205.07848/) (2022)
- [BEINIT: Avoiding Barren Plateaus in Variational Quantum Algorithms](https://mlsanity.com/paper/2204.13751/) (2022)
- [Concentration of Data Encoding in Parameterized Quantum Circuits](https://mlsanity.com/paper/2206.08273/) (2022)
- [Graph neural network initialisation of quantum approximate optimisation](https://mlsanity.com/paper/2111.03016/) (2021)
- [Variational Quantum Singular Value Decomposition](https://mlsanity.com/paper/2006.02336/) (2020)
- [Effects of noise on the overparametrization of quantum neural networks](https://mlsanity.com/paper/2302.05059/) (2023)
- [Mitigating barren plateaus of variational quantum eigensolvers](https://mlsanity.com/paper/2205.13539/) (2022)
- [Analyzing Convergence in Quantum Neural Networks: Deviations from Neural Tangent Kernels](https://mlsanity.com/paper/2303.14844/) (2023)
- [Variational quantum Gibbs state preparation with a truncated Taylor series](https://mlsanity.com/paper/2005.08797/) (2020)
- [Sublinear classical and quantum algorithms for general matrix games](https://mlsanity.com/paper/2012.06519/) (2020)

## Built on by (most cited first, 1 of 1)

- [Magic-Informed Quantum Architecture Search](https://mlsanity.com/paper/2605.03932/) (2026)

## Abstract (arXiv)

Quantum neural networks (QNNs) have been a promising framework in pursuing near-term quantum advantage in various fields, where many applications can be viewed as learning a quantum state that encodes useful data. As a quantum analog of probability distribution learning, quantum state learning is theoretically and practically essential in quantum machine learning. In this paper, we develop a no-go theorem for learning an unknown quantum state with QNNs even starting from a high-fidelity initial state. We prove that when the loss value is lower than a critical threshold, the probability of avoiding local minima vanishes exponentially with the qubit count, while only grows polynomially with the circuit depth. The curvature of local minima is concentrated to the quantum Fisher information times a loss-dependent constant, which characterizes the sensibility of the output state with respect to parameters in QNNs. These results hold for any circuit structures, initialization strategies, and work for both fixed ansatzes and adaptive methods. Extensive numerical simulations are performed to validate our theoretical results. Our findings place generic limits on good initial guesses and adaptive methods for improving the learnability and scalability of QNNs, and deepen the understanding of prior information's role in QNNs.