# Non-variational supervised quantum kernel methods: a review

arXiv: https://arxiv.org/abs/2604.07896 · HTML: https://mlsanity.com/paper/2604.07896/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 39 of its 119 references that are in the map.
- First posted on arXiv 9 Apr 2026.

## Builds on (most cited first, 30 of 39)

- [Neural Tangent Kernel: Convergence and Generalization in Neural Networks](https://mlsanity.com/paper/1806.07572/) (2018)
- [Variational Quantum Algorithms](https://mlsanity.com/paper/2012.09265/) (2020)
- [Supervised learning with quantum enhanced feature spaces](https://mlsanity.com/paper/1804.11326/) (2018)
- [Power of data in quantum machine learning](https://mlsanity.com/paper/2011.01938/) (2020)
- [The effect of data encoding on the expressive power of variational quantum machine learning models](https://mlsanity.com/paper/2008.08605/) (2020)
- [Parameterized quantum circuits as machine learning models](https://mlsanity.com/paper/1906.07682/) (2019)
- [A rigorous and robust quantum speed-up in supervised machine learning](https://mlsanity.com/paper/2010.02174/) (2020)
- [Predicting Many Properties of a Quantum System from Very Few Measurements](https://mlsanity.com/paper/2002.08953/) (2020)
- [Supervised quantum machine learning models are kernel methods](https://mlsanity.com/paper/2101.11020/) (2021)
- [Noise-Induced Barren Plateaus in Variational Quantum Algorithms](https://mlsanity.com/paper/2007.14384/) (2020)
- [Connecting ansatz expressibility to gradient magnitudes and barren plateaus](https://mlsanity.com/paper/2101.02138/) (2021)
- [Spectral Bias and Task-Model Alignment Explain Generalization in Kernel Regression and Infinitely Wide Neural Networks](https://mlsanity.com/paper/2006.13198/) (2020)
- [Efficient Learning for Deep Quantum Neural Networks](https://mlsanity.com/paper/1902.10445/) (2019)
- [Barren Plateaus in Variational Quantum Computing](https://mlsanity.com/paper/2405.00781/) (2024)
- [Absence of Barren Plateaus in Quantum Convolutional Neural Networks](https://mlsanity.com/paper/2011.02966/) (2020)
- [Entanglement Induced Barren Plateaus](https://mlsanity.com/paper/2010.15968/) (2020)
- [Quantum machine learning beyond kernel methods](https://mlsanity.com/paper/2110.13162/) (2021)
- [The Inductive Bias of Quantum Kernels](https://mlsanity.com/paper/2106.03747/) (2021)
- [Exponential concentration in quantum kernel methods](https://mlsanity.com/paper/2208.11060/) (2022)
- [Better than classical? The subtle art of benchmarking quantum machine learning models](https://mlsanity.com/paper/2403.07059/) (2024)
- [Does provable absence of barren plateaus imply classical simulability?](https://mlsanity.com/paper/2312.09121/) (2023)
- [Towards understanding the power of quantum kernels in the NISQ era](https://mlsanity.com/paper/2103.16774/) (2021)
- [Representation Learning via Quantum Neural Tangent Kernels](https://mlsanity.com/paper/2111.04225/) (2021)
- [Importance of Kernel Bandwidth in Quantum Machine Learning](https://mlsanity.com/paper/2111.05451/) (2021)
- [Automatic design of quantum feature maps](https://mlsanity.com/paper/2105.12626/) (2021)
- [Potential and limitations of random Fourier features for dequantizing quantum machine learning](https://mlsanity.com/paper/2309.11647/) (2023)
- [Quantum Kernel Methods under Scrutiny: A Benchmarking Study](https://mlsanity.com/paper/2409.04406/) (2024)
- [Quantum assisted Gaussian process regression](https://mlsanity.com/paper/1512.03929/) (2015)
- [Quantum kernels for real-world predictions based on electronic health records](https://mlsanity.com/paper/2112.06211/) (2021)
- [Detecting Clouds in Multispectral Satellite Images Using Quantum-Kernel Support Vector Machines](https://mlsanity.com/paper/2302.08270/) (2023)

## Built on by (most cited first, 2 of 2)

- [Optimal algorithmic complexity of inference in quantum kernel methods](https://mlsanity.com/paper/2604.15214/) (2026)
- [Quantum Topological Data Encoding](https://mlsanity.com/paper/2607.13847/) (2026)

## Abstract (arXiv)

Quantum kernel methods (QKMs) have emerged as a prominent framework for supervised quantum machine learning. Unlike variational quantum algorithms, which rely on gradient-based optimisation and may suffer from issues such as barren plateaus, non-variational QKMs employ fixed quantum feature maps, with model selection performed classically via convex optimisation and cross-validation. This separation of quantum feature embedding from classical training ensures stable optimisation while leveraging quantum circuits to encode data in high-dimensional Hilbert spaces. In this review, we provide a thorough analysis of non-variational supervised QKMs, covering their foundations in classical kernel theory, constructions of fidelity and projected quantum kernels, and methods for their estimation in practice. We examine frameworks for assessing quantum advantage, including generalisation bounds and necessary conditions for separation from classical models, and analyse key challenges such as exponential concentration, dequantisation via tensor-network methods, and the spectral properties of kernel integral operators. We further discuss structured problem classes that may enable advantage, and synthesise insights from comparative and hardware studies. Overall, this review aims to clarify the regimes in which QKMs may offer genuine advantages, and to delineate the conceptual, methodological, and technical obstacles that must be overcome for practical quantum-enhanced learning.