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The popular qubit framework has dominated recent work on quantum kernels,with results characterising expressability, learnability and generalisation. Asyet, there is no comparative framework to understand these concepts forcontinuous variable (CV) quantum computing platforms. In this paper werepresent CV quantum kernels as holomorphic functions and use thisrepresentation to provide several important theoretical insights. The approachpermits a general closed form solution for all CV quantum kernels and showsevery such kernel can be expressed as the product of Gaussian and polynomialterms. Furthermore, it enables quantification of a quantum-classical separationfor all such kernels via a notion of "stellar rank", and provides intuition forhow bandwidth hyper-parameter tuning results in trades-off between learnabilityand efficient classical simulability.

A General Form for Continuous Variable Quantum Kernels