Berlin 2024 – wissenschaftliches Programm
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QI: Fachverband Quanteninformation
QI 9: Quantum Machine Learning and Classical Simulability
QI 9.8: Vortrag
Dienstag, 19. März 2024, 11:45–12:00, HFT-FT 101
On the expressivity of embedding quantum kernels — •Elies Gil-Fuster1,2, Jens Eisert1,2,3, and Vedran Dunjko4,5 — 1Dahlem Center for Complex Quantum Systems, Freie Universitat Berlin — 2Fraunhofer Heinrich Hertz Institute, Berlin — 3Helmholtz-Zentrum Berlin fur Materialien und Energie — 4Applied Quantum Algorithms, Universiteit Leiden, Netherlands — 5LIACS, Universiteit Leiden, Netherlands
One of the most natural connections between quantum and classical machine learning has been established in the context of kernel methods. Quantum kernels are typically evaluated by explicitly constructing quantum feature states and then taking their inner product, here called embedding quantum kernels. Since classical kernels are usually evaluated without using the feature vectors explicitly, we wonder how expressive embedding quantum kernels are. In this work, we raise the question: can all quantum kernels be expressed as the inner product of quantum feature states? Our first result is positive: for any kernel function there always exists a corresponding quantum feature map and an embedding quantum kernel. In a second part, we formalize the question of universality of efficient embedding quantum kernels. We show that efficient embedding quantum kernels are universal within a broad class of shift invariant kernels. We then extend this result to a new class of so-called composition kernels, which we show also contains projected quantum kernels introduced in recent works. We finally identify the directions towards new, more exotic, and unexplored quantum kernel families.
Keywords: expressivity; learning theory; quantum kernels; kernel methods