# Position: Quantum Kernel Machines Should Move Beyond Scalar-Valued Kernels to Realize Their Potential

arXiv: https://arxiv.org/abs/2506.03779 · HTML: https://mlsanity.com/paper/2506.03779/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 44 of its 102 references that are in the map.
- First posted on arXiv 4 Jun 2025.

## Builds on (most cited first, 30 of 44)

- [Reconciling modern machine learning practice and the bias-variance trade-off](https://mlsanity.com/paper/1812.11118/) (2018)
- [Link Prediction Based on Graph Neural Networks](https://mlsanity.com/paper/1802.09691/) (2018)
- [Supervised learning with quantum enhanced feature spaces](https://mlsanity.com/paper/1804.11326/) (2018)
- [Benign Overfitting in Linear Regression](https://mlsanity.com/paper/1906.11300/) (2019)
- [Power of data in quantum machine learning](https://mlsanity.com/paper/2011.01938/) (2020)
- [A rigorous and robust quantum speed-up in supervised machine learning](https://mlsanity.com/paper/2010.02174/) (2020)
- [Supervised quantum machine learning models are kernel methods](https://mlsanity.com/paper/2101.11020/) (2021)
- [Random Features for Kernel Approximation: A Survey on Algorithms, Theory, and Beyond](https://mlsanity.com/paper/2004.11154/) (2020)
- [Theory of overparametrization in quantum neural networks](https://mlsanity.com/paper/2109.11676/) (2021)
- [Quantum machine learning: a classical perspective](https://mlsanity.com/paper/1707.08561/) (2017)
- [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)
- [A Graph to Graphs Framework for Retrosynthesis Prediction](https://mlsanity.com/paper/2003.12725/) (2020)
- [The Born Supremacy: Quantum Advantage and Training of an Ising Born Machine](https://mlsanity.com/paper/1904.02214/) (2019)
- [Training Quantum Embedding Kernels on Near-Term Quantum Computers](https://mlsanity.com/paper/2105.02276/) (2021)
- [Graph Kernels: State-of-the-Art and Future Challenges](https://mlsanity.com/paper/2011.03854/) (2020)
- [Operator-valued Kernels for Learning from Functional Response Data](https://mlsanity.com/paper/1510.08231/) (2015)
- [Understanding quantum machine learning also requires rethinking generalization](https://mlsanity.com/paper/2306.13461/) (2023)
- [Towards understanding the power of quantum kernels in the NISQ era](https://mlsanity.com/paper/2103.16774/) (2021)
- [Challenges and Opportunities in Quantum Optimization](https://mlsanity.com/paper/2312.02279/) (2023)
- [Bandwidth Enables Generalization in Quantum Kernel Models](https://mlsanity.com/paper/2206.06686/) (2022)
- [Quantum Computing: Lecture Notes](https://mlsanity.com/paper/1907.09415/) (2019)
- [Quantum Kernel Alignment with Stochastic Gradient Descent](https://mlsanity.com/paper/2304.09899/) (2023)
- [Scalable Matrix-valued Kernel Learning for High-dimensional Nonlinear Multivariate Regression and Granger Causality](https://mlsanity.com/paper/1408.2066/) (2014)
- [Quantum Advantage Seeker with Kernels (QuASK): a software framework to speed up the research in quantum machine learning](https://mlsanity.com/paper/2206.15284/) (2022)
- [Reproducing kernel Hilbert C*-module and kernel mean embeddings](https://mlsanity.com/paper/2101.11410/) (2021)
- [Automatic and effective discovery of quantum kernels](https://mlsanity.com/paper/2209.11144/) (2022)
- [Double descent in quantum kernel methods](https://mlsanity.com/paper/2501.10077/) (2025)
- [Towards a theory of non-commutative optimization: geodesic first and second order methods for moment maps and polytopes](https://mlsanity.com/paper/1910.12375/) (2019)

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

- [When AI meets quantum information: A comprehensive review](https://mlsanity.com/paper/2607.00365/) (2026)

## Abstract (arXiv)

Quantum kernel functions built using quantum-mechanical principles and have emerged as a centerpiece of quantum machine learning. The initial enthusiasm for quantum kernel machines has been tempered by recent studies suggesting that quantum kernels could not offer significant computational or statistical advantages when learning from classical data. However, most of the research in this area has been devoted to scalar-valued kernels in standard classification or regression settings for which classical kernel methods are efficient and effective, leaving very little room for improvement with quantum kernels. In this position paper, we argue that progress in this field requires moving beyond scalar-valued kernels toward more expressive kernel frameworks. Scalar-valued kernels lack the degrees of freedom necessary to fully exploit intrinsically quantum resources such as entanglement and are not rich enough to deal with complex learning tasks where classical learning methods struggle. Building on recent advances in operator-valued kernel learning and C^*-algebraic kernel representations, we propose a roadmap for designing quantum kernels capable of leveraging entanglement and non-commutative structures to tackle complex structured prediction problems. To support this viewpoint, we present an initial proof-of-concept illustrating how quantum operator-valued kernel formulations can reveal structural dependencies that remain difficult to access for scalar-valued kernel methods. This shift in focus could open a pathway toward a new generation of quantum kernel machines and a more faithful exploration of their potential advantages.