Publication record · 18.cifr/2019.schuld.quantum-kernel-hilbert

Quantum Machine Learning in Feature Hilbert Spaces

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Maria Schuld (Xanadu and University of KwaZulu-Natal), Nathan Killoran (Xanadu)

RAI18.cifr/2019.schuld.quantum-kernel-hilbert

Physical Review Letters· 2019· doi:10.1103/PhysRevLett.122.040504

We connect quantum machine learning to the framework of statistical learning theory via the notion of a quantum feature map. In particular, we show that quantum computers can be used to implement kernel functions on a Hilbert space of quantum states, and that kernel-based classification algorithms can be executed efficiently by estimating the kernel with a quantum device and classifying with a support vector machine.

quantum machine learningkernel methodsfeature mapsupport vector machinequantum kernelHilbert space

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Quantum machine learning lacked a principled theoretical foundation linking quantum circuits to classical learning theory. Practitioners designing variational circuits had no framework to reason about expressivity or generalization. This work fills that gap by identifying the RKHS induced by a quantum feature map, giving a complete kernel-method interpretation.

Novelty

This paper establishes that quantum feature maps implicitly define kernel functions in a Hilbert space, reframing quantum machine learning as kernel SVM with a quantum-computed kernel. It provides the first rigorous theoretical bridge between variational quantum circuits and reproducing kernel Hilbert space (RKHS) theory, enabling QML to leverage classical kernel-method guarantees.

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