# Universal Approximation Property of Quantum Machine Learning Models in Quantum-Enhanced Feature Spaces

arXiv: https://arxiv.org/abs/2009.00298 · HTML: https://mlsanity.com/paper/2009.00298/

## Key facts

- Cited by 44 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 1 Sep 2020.

## Built on by (most cited first, 30 of 44)

- [Quantum machine learning beyond kernel methods](https://mlsanity.com/paper/2110.13162/) (2021)
- [Parametrized quantum policies for reinforcement learning](https://mlsanity.com/paper/2103.05577/) (2021)
- [QKSAN: A Quantum Kernel Self-Attention Network](https://mlsanity.com/paper/2308.13422/) (2023)
- [Quantum Policy Gradient Algorithm with Optimized Action Decoding](https://mlsanity.com/paper/2212.06663/) (2022)
- [Style-based quantum generative adversarial networks for Monte Carlo events](https://mlsanity.com/paper/2110.06933/) (2021)
- [Power and limitations of single-qubit native quantum neural networks](https://mlsanity.com/paper/2205.07848/) (2022)
- [On fundamental aspects of quantum extreme learning machines](https://mlsanity.com/paper/2312.15124/) (2023)
- [Universal Approximation Theorem and error bounds for quantum neural networks and quantum reservoirs](https://mlsanity.com/paper/2307.12904/) (2023)
- [Reduction of finite sampling noise in quantum neural networks](https://mlsanity.com/paper/2306.01639/) (2023)
- [Fock State-enhanced Expressivity of Quantum Machine Learning Models](https://mlsanity.com/paper/2107.05224/) (2021)
- [Non-asymptotic Approximation Error Bounds of Parameterized Quantum Circuits](https://mlsanity.com/paper/2310.07528/) (2023)
- [QCPINN: Quantum-Classical Physics-Informed Neural Networks for Solving PDEs](https://mlsanity.com/paper/2503.16678/) (2025)
- [Expressive variational quantum circuits provide inherent privacy in federated learning](https://mlsanity.com/paper/2309.13002/) (2023)
- [Unleashing the Expressive Power of Pulse-Based Quantum Neural Networks](https://mlsanity.com/paper/2402.02880/) (2024)
- [Efficient Quantum Feature Extraction for CNN-based Learning](https://mlsanity.com/paper/2201.01246/) (2022)
- [Provable advantages of kernel-based quantum learners and quantum preprocessing based on Grover's algorithm](https://mlsanity.com/paper/2309.14406/) (2023)
- [Guided Quantum Compression for High Dimensional Data Classification](https://mlsanity.com/paper/2402.09524/) (2024)
- [Iterative Quantum Feature Maps](https://mlsanity.com/paper/2506.19461/) (2025)
- [Equivariant Quantum Graph Circuits](https://mlsanity.com/paper/2112.05261/) (2021)
- [The Impact of Feature Embedding Placement in the Ansatz of a Quantum Kernel in QSVMs](https://mlsanity.com/paper/2409.13147/) (2024)
- [QuXAI: Explainers for Hybrid Quantum Machine Learning Models](https://mlsanity.com/paper/2505.10167/) (2025)
- [Harmonic (Quantum) Neural Networks](https://mlsanity.com/paper/2212.07462/) (2022)
- [Pre-training Tensor-Train Networks Facilitates Machine Learning with Variational Quantum Circuits](https://mlsanity.com/paper/2306.03741/) (2023)
- [Streaming IoT Data and the Quantum Edge: A Classic/Quantum Machine Learning Use Case](https://mlsanity.com/paper/2402.15542/) (2024)
- [On Optimizing Hyperparameters for Quantum Neural Networks](https://mlsanity.com/paper/2403.18579/) (2024)
- [Quantum Curriculum Learning](https://mlsanity.com/paper/2407.02419/) (2024)
- [Vision-QRWKV: Exploring Quantum-Enhanced RWKV Models for Image Classification](https://mlsanity.com/paper/2506.06633/) (2025)
- [Approximation rates of quantum neural networks for periodic functions via Jackson's inequality](https://mlsanity.com/paper/2511.16149/) (2025)
- [Quantum-Enhanced Neural Contextual Bandit Algorithms](https://mlsanity.com/paper/2601.02870/) (2026)
- [SAQNN: Spectral Adaptive Quantum Neural Network as a Universal Approximator](https://mlsanity.com/paper/2602.09718/) (2026)

## Abstract (arXiv)

Encoding classical data into quantum states is considered a quantum feature map to map classical data into a quantum Hilbert space. This feature map provides opportunities to incorporate quantum advantages into machine learning algorithms to be performed on near-term intermediate-scale quantum computers. The crucial idea is using the quantum Hilbert space as a quantum-enhanced feature space in machine learning models. While the quantum feature map has demonstrated its capability when combined with linear classification models in some specific applications, its expressive power from the theoretical perspective remains unknown. We prove that the machine learning models induced from the quantum-enhanced feature space are universal approximators of continuous functions under typical quantum feature maps. We also study the capability of quantum feature maps in the classification of disjoint regions. Our work enables an important theoretical analysis to ensure that machine learning algorithms based on quantum feature maps can handle a broad class of machine learning tasks. In light of this, one can design a quantum machine learning model with more powerful expressivity.