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In this paper we represent CV quantum kernels as closed form functions and use this representation to provide several important theoretical insights. We derive a general closed form solution for all CV quantum kernels and show every such kernel can be expressed as the product of a Gaussian and an algebraic function of the parameters of the feature map. Furthermore, in the multi-mode case, we present quantification of a quantum-classical separation for all quantum kernels via a hierarchical notion of the \u201cstellar rank\" of the quantum kernel feature map. We then prove kernels defined by feature maps of infinite stellar rank, such as GKP-state encodings, can be approximated arbitrarily well by kernels defined by feature maps of finite stellar rank. Finally, we simulate learning with a single-mode displaced Fock state encoding and show that (i) accuracy on our specific task (an annular data set) increases with stellar rank, (ii) for underfit models, accuracy can be improved by increasing a bandwidth hyperparameter, and (iii) for noisy data that is overfit, decreasing the bandwidth will improve generalisation but does so at the cost of effective stellar rank.<\/jats:p>","DOI":"10.22331\/q-2024-12-17-1570","type":"journal-article","created":{"date-parts":[[2024,12,17]],"date-time":"2024-12-17T15:26:36Z","timestamp":1734449196000},"page":"1570","update-policy":"https:\/\/doi.org\/10.22331\/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":5,"title":["Quantum Kernel Machine Learning With Continuous Variables"],"prefix":"10.22331","volume":"8","author":[{"given":"Laura J.","family":"Henderson","sequence":"first","affiliation":[{"name":"School of Mathematics and Physics, The University of Queensland, QLD 4072, Australia"},{"name":"ARC Centre for Engineered Quantum Systems, The University of Queensland, QLD, 4072, Australia."}]},{"given":"Rishi","family":"Goel","sequence":"additional","affiliation":[{"name":"School of Mathematics and Physics, The University of Queensland, QLD 4072, Australia"}]},{"given":"Sally","family":"Shrapnel","sequence":"additional","affiliation":[{"name":"School of Mathematics and Physics, The University of Queensland, QLD 4072, Australia"},{"name":"ARC Centre for Engineered Quantum Systems, The University of Queensland, QLD, 4072, Australia."}]}],"member":"9598","published-online":{"date-parts":[[2024,12,17]]},"reference":[{"key":"0","doi-asserted-by":"publisher","unstructured":"Maria Schuld and Francesco Petruccione. ``Quantum models as kernel methods''. 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