  Quantum computing algorithms have been shown to produce performant quantum
kernels for machine-learning classification problems. Here, we examine the
performance of quantum kernels for regression problems of practical interest.
For an unbiased benchmarking of quantum kernels, it is necessary to construct
the most optimal functional form of the classical kernels and the most optimal
quantum kernels for each given data set. We develop an algorithm that uses an
analog of the Bayesian information criterion to optimize the sequence of
quantum gates used to estimate quantum kernels for Gaussian process models. The
algorithm increases the complexity of the quantum circuits incrementally, while
improving the performance of the resulting kernels, and is shown to yield much
higher model accuracy with fewer quantum gates than a fixed quantum circuit
ansatz. We demonstrate that quantum kernels thus obtained can be used to build
accurate models of global potential energy surfaces (PES) for polyatomic
molecules. The average interpolation error of the six-dimensional PES obtained
with a random distribution of 2000 energy points is 16 cm$^{-1}$ for
H$_3$O$^+$, 15 cm$^{-1}$ for H$_2$CO and 88 cm$^{-1}$ for HNO$_2$. We show that
a compositional optimization of classical kernels for Gaussian process
regression converges to the same errors. This indicates that quantum kernels
can achieve the same, though not better, expressivity as classical kernels for
regression problems.
