Quantum Algorithms for Training Gaussian Processes - Oxford Man Institute of Quantitative Finance 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. The LML gives a concrete measure of the quality of prediction that a GP model is expected to achieve. The classical computation of the 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 yTA−1y, wherey 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.Oxford Man Institute of Quantitative Finance