  In this thesis, I present several results on quantum statistical inference in
the following two directions. Firstly, I demonstrate that quantum algorithms
can be applied to enhance the computing and training of Gaussian processes
(GPs), a powerful model widely used in classical statistical inference and
supervised machine learning. A crucial component of the quantum GP algorithm is
solving linear systems with quantum computers, for which I present a novel
algorithm that achieves a provable advantage over previously known methods. I
will also explicitly address the task of encoding the classical data into a
quantum state for machine learning applications. I then apply the quantum
enhanced GPs to Bayesian deep learning and present an experimental
demonstration on contemporary hardware and simulators. Secondly, I look into
the notion of quantum causality and apply it to inferring spatial and temporal
quantum correlations, and present an analytical toolkit for causal inference in
quantum data. I will also make the connection between causality and quantum
communications, and present a general bound for the quantum capacity of noisy
communication channels.
