  The kernel mean embedding of probability distributions is commonly used in
machine learning as an injective mapping from distributions to functions in an
infinite dimensional Hilbert space. It allows us, for example, to define a
distance measure between probability distributions, called maximum mean
discrepancy (MMD). In this work, we propose to represent probability
distributions in a pure quantum state of a system that is described by an
infinite dimensional Hilbert space. This enables us to work with an explicit
representation of the mean embedding, whereas classically one can only work
implicitly with an infinite dimensional Hilbert space through the use of the
kernel trick. We show how this explicit representation can speed up methods
that rely on inner products of mean embeddings and discuss the theoretical and
experimental challenges that need to be solved in order to achieve these
speedups.
