Quantum Predictive Learning and Communication Complexity with Single Input

We define a new model of quantum learning that we call Predictive Quantum (PQ). This is a quantum analogue of PAC, where during the testing phase the student is only required to answer a polynomial number of testing queries. We demonstrate a relational concept class that is efficiently learnable in PQ, while in any reasonable classical model exponential amount of training data would be required. This is the first unconditional separation between quantum and classical learning. We show that our separation is the best possible in several ways; in particular, there is no analogous result for a functional class, as well as for several weak