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We introduce a general statistical learning theory for processes that take asinput a classical random variable and output a quantum state. Our setting ismotivated by the practical situation in which one desires to learn a quantumprocess governed by classical parameters that are out of one's control, e.g.,in learning astronomical processes, studying disordered systems and biologicalprocesses not controlled by the observer. We provide an algorithm for learningwith high probability in this setting with a finite amount of samples, even ifthe concept class is infinite. To do this, we review and adapt existingalgorithms for shadow tomography and hypothesis selection, and combine theirguarantees with the uniform convergence on the data of the loss functions ofinterest. As a by-product we obtain sufficient conditions for performing shadowtomography of classical-quantum states with a number of copies which depend onthe dimension of the quantum register, but not on the dimension of theclassical one. We give concrete examples of processes that can be learned inthis manner, based on quantum circuits or physically motivated scenarios likesystems subject to Hamiltonians with random perturbations or data-dependentphase-shifts.

Learning quantum processes without input control