  Gaussian processes (GPs) are important models in supervised machine learning.
Training in Gaussian processes refers to selecting the covariance functions and
the associated parameters in order to improve the outcome of predictions, the
core of which amounts to evaluating the logarithm of the marginal likelihood
(LML) of a given model. LML gives a concrete measure of the quality of
prediction that a GP model is expected to achieve. The classical computation of
LML typically carries a polynomial time overhead with respect to the input
size. We propose a quantum algorithm that computes the logarithm of the
determinant of a Hermitian matrix, which runs in logarithmic time for sparse
matrices. This is applied in conjunction with a variant of the quantum linear
system algorithm that allows for logarithmic time computation of the form
$\mathbf{y}^TA^{-1}\mathbf{y}$, where $\mathbf{y}$ is a dense vector and $A$ is
the covariance matrix. We hence show that quantum computing can be used to
estimate the LML of a GP with exponentially improved efficiency under certain
conditions.
