With gates of a quantum computer designed to encode multi-dimensional vectors, projections of quantum computer states onto specific qubit states can produce kernels of a reproducing kernel Hilbert space. This presentation will target the following question: Can quantum kernels outperform classical kernels for supervised learning problems, when observations are exceedingly expensive? After a brief introduction of reproducing kernel Hilbert spaces, I will discuss our work on building performant kernels for extrapolation and optimization problems. Motivated by this discussion, I will then present our recent algorithms for building optimal quantum kernels for regression and classification problems.