Quantum State Classification via Quantum Fourier

We study learning from quantum data, in particular quantum state classification which has applications, among others, in classifying the separability of quantum states. In this learning model, there are $n$ quantum states with classical labels as the training samples. Predictors are quantum measurements that when applied to the next unseen quantum state predict its classical label. By integrating learning theory with quantum information, we introduce a quantum counterpart of the PAC framework for learning with respect to classes of measurements. We argue that major challenges arising from the quantum nature of the problem are measurement incompatibility and the no-cloning principle -- prohibiting sample reuse. Then, after introducing a Fourier expansion through Pauli's operators, we study learning with respect to an infinite class of quantum measurements whose operator's Fourier spectrum is concentrated on low degree terms. We propose a quantum learning algorithm and show that the quantum sample complexity depends on the ``compatibility structure" of such measurement classes -- the more compatible the class is, the lower the quantum sample complexity will be. We further introduce $k$-junta measurements as a special class of low-depth quantum circuits whose Fourier spectrum is concentrated on low degrees.