The problem of private learning has been extensively studied in classical computer science. Notably, a striking equivalence between local differentially private learning and statistical query learning has been shown. In addition, the statistical query model has been recently extended to quantum computation. In this work, we give a formal definition of quantum local differential privacy and we extend the aforementioned result to quantum computation.

Information-theoretic lower bounds are often encountered in several branches of computer science, including learning theory and cryptography. In the quantum setting, Holevo's and Nayak's bounds give an estimate of the amount of classical information that can be stored in a quantum state. Previous works have shown how to combine information-theoretic tools with a counting argument to lower bound the sample complexity of distribution-free quantum probably approximately correct (PAC) learning. In our work, we establish the notion of Probably Approximately Correct Source Coding and we show two novel applications in quantum learning theory and delegated quantum computation with a purely classical client. In particular, we provide a lower bound of the sample complexity of a quantum learner for arbitrary functions under the Zipf distribution, and we improve the security guarantees of a classically-driven delegation protocol for measurement-based quantum computation (MBQC).