# Power and limitations of single-qubit native quantum neural networks

arXiv: https://arxiv.org/abs/2205.07848 · HTML: https://mlsanity.com/paper/2205.07848/

## Key facts

- Cited by 19 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 58 references that are in the map.
- First posted on arXiv 16 May 2022.

## Builds on (most cited first, 9 of 9)

- [Variational Quantum Algorithms](https://mlsanity.com/paper/2012.09265/) (2020)
- [The effect of data encoding on the expressive power of variational quantum machine learning models](https://mlsanity.com/paper/2008.08605/) (2020)
- [Quantum machine learning beyond kernel methods](https://mlsanity.com/paper/2110.13162/) (2021)
- [Noisy intermediate-scale quantum (NISQ) algorithms](https://mlsanity.com/paper/2101.08448/) (2021)
- [Out-of-distribution generalization for learning quantum dynamics](https://mlsanity.com/paper/2204.10268/) (2022)
- [Universal Approximation Property of Quantum Machine Learning Models in Quantum-Enhanced Feature Spaces](https://mlsanity.com/paper/2009.00298/) (2020)
- [Recent advances for quantum classifiers](https://mlsanity.com/paper/2108.13421/) (2021)
- [VSQL: Variational Shadow Quantum Learning for Classification](https://mlsanity.com/paper/2012.08288/) (2020)
- [Optimal quantum dataset for learning a unitary transformation](https://mlsanity.com/paper/2203.00546/) (2022)

## Built on by (most cited first, 19 of 19)

- [On the relation between trainability and dequantization of variational quantum learning models](https://mlsanity.com/paper/2406.07072/) (2024)
- [Quantum Implicit Neural Representations](https://mlsanity.com/paper/2406.03873/) (2024)
- [Non-asymptotic Approximation Error Bounds of Parameterized Quantum Circuits](https://mlsanity.com/paper/2310.07528/) (2023)
- [Quantum Variational Activation Functions Empower Kolmogorov-Arnold Networks](https://mlsanity.com/paper/2509.14026/) (2025)
- [QKAN-LSTM: Quantum-inspired Kolmogorov-Arnold Long Short-term Memory](https://mlsanity.com/paper/2512.05049/) (2025)
- [Unleashing the Expressive Power of Pulse-Based Quantum Neural Networks](https://mlsanity.com/paper/2402.02880/) (2024)
- [Predictive Performance of Deep Quantum Data Re-uploading Models](https://mlsanity.com/paper/2505.20337/) (2025)
- [Regression and Classification with Single-Qubit Quantum Neural Networks](https://mlsanity.com/paper/2412.09486/) (2024)
- [Statistical Analysis of Quantum State Learning Process in Quantum Neural Networks](https://mlsanity.com/paper/2309.14980/) (2023)
- [Parameterized quantum comb and simpler circuits for reversing unknown qubit-unitary operations](https://mlsanity.com/paper/2403.03761/) (2024)
- [Hybrid Quantum Neural Networks for Efficient Protein-Ligand Binding Affinity Prediction](https://mlsanity.com/paper/2509.11046/) (2025)
- [Approximation rates of quantum neural networks for periodic functions via Jackson's inequality](https://mlsanity.com/paper/2511.16149/) (2025)
- [SAQNN: Spectral Adaptive Quantum Neural Network as a Universal Approximator](https://mlsanity.com/paper/2602.09718/) (2026)
- [A Simple Quantum Blockmodeling with Qubits and Permutations](https://mlsanity.com/paper/2311.07726/) (2023)
- [Single-Qudit Quantum Neural Networks for Multiclass Classification](https://mlsanity.com/paper/2503.09269/) (2025)
- [On the Design of Expressive and Trainable Pulse-based Quantum Machine Learning Models](https://mlsanity.com/paper/2508.05559/) (2025)
- [Quantum Visual Word Sense Disambiguation: Unraveling Ambiguities Through Quantum Inference Model](https://mlsanity.com/paper/2512.24687/) (2025)
- [Trainability-Oriented Hybrid Quantum Regression via Geometric Preconditioning and Curriculum Optimization](https://mlsanity.com/paper/2601.11942/) (2026)
- [Long Range Frequency Tuning for QML](https://mlsanity.com/paper/2602.23409/) (2026)

## Abstract (arXiv)

Quantum neural networks (QNNs) have emerged as a leading strategy to establish applications in machine learning, chemistry, and optimization. While the applications of QNN have been widely investigated, its theoretical foundation remains less understood. In this paper, we formulate a theoretical framework for the expressive ability of data re-uploading quantum neural networks that consist of interleaved encoding circuit blocks and trainable circuit blocks. First, we prove that single-qubit quantum neural networks can approximate any univariate function by mapping the model to a partial Fourier series. We in particular establish the exact correlations between the parameters of the trainable gates and the Fourier coefficients, resolving an open problem on the universal approximation property of QNN. Second, we discuss the limitations of single-qubit native QNNs on approximating multivariate functions by analyzing the frequency spectrum and the flexibility of Fourier coefficients. We further demonstrate the expressivity and limitations of single-qubit native QNNs via numerical experiments. We believe these results would improve our understanding of QNNs and provide a helpful guideline for designing powerful QNNs for machine learning tasks.