  We introduce a general statistical learning theory for processes that take as
input a classical random variable and output a quantum state. Our setting is
motivated by the practical situation in which one desires to learn a quantum
process governed by classical parameters that are out of one's control. This
framework is applicable, for example, to the study of astronomical phenomena,
disordered systems and biological processes not controlled by the observer. We
provide an algorithm for learning with high probability in this setting with a
finite amount of samples, even if the concept class is infinite. To do this, we
review and adapt existing algorithms for shadow tomography and hypothesis
selection, and combine their guarantees with the uniform convergence on the
data of the loss functions of interest. As a by-product we obtain sufficient
conditions for performing shadow tomography of classical-quantum states with a
number of copies which depends on the dimension of the quantum register, but
not on the dimension of the classical one. We give concrete examples of
processes that can be learned in this manner, based on quantum circuits or
physically motivated classes, such as systems governed by Hamiltonians with
random perturbations or data-dependent phase-shifts.
