A Theoretical Framework for Learning from Quantum Data

Over decades traditional information theory of source and channel coding advances toward learning and effective extraction of information from data. We propose to go one step further and offer a theoretical foundation for learning classical patterns from quantum data. However, there are several roadblocks to lay the groundwork for such a generalization. First, classical data must be replaced by a density operator over a Hilbert space. Hence, deviated from problems such as state tomography, our samples are i.i.d density operators. The second challenge is even more profound since we must realize that our only interaction with a quantum state is through a measurement which - due to no-cloning quantum postulate - loses information after measuring it. With this in mind, we present a quantum counterpart of the well-known probably approximately correct (PAC) framework. Based on that, we propose a quantum analogous of the Empirical Risk Minimization (ERM) algorithm for learning measurement hypothesis classes. Then, we establish upper bounds on the quantum sample complexity quantum concept classes.