New Bounds on Quantum Sample Complexity of Measurement Classes

This paper studies quantum supervised learning for classical inference from quantum states. In this model, a learner has access to a set of labeled quantum samples as the training set. The objective is to find a quantum measurement that predicts the label of the unseen samples. The hardness of learning is measured via sample complexity under a quantum counterpart of the well-known probably approximately correct (PAC). Quantum sample complexity is expected to be higher than classical one, because of the measurement incompatibility and state collapse. Recent efforts showed that the sample complexity of learning a finite quantum concept class $\mathcal{C}$ scales as $O(\vert \mathcal{C}\vert)$. This is significantly higher than the classical sample complexity that grows logarithmically with the class size. This work improves the sample complexity bound to $O\left(V_{\mathcal{C}^*} \log \left\vert\mathcal{C}^*\right\vert\right)$, where $\mathcal{C}^*$ is the set of extreme points of the convex closure of $\mathcal{C}$ and $V_{\mathcal{C}^*}$ is the shadow-norm of this set. We show the tightness of our bound for the class of bounded Hilbert-Schmidt norm, scaling as $O\left(\log \left\vert\mathcal{C}^*\right\vert\right)$. Our approach is based on a new quantum empirical risk minimization (ERM) algorithm equipped with a shadow tomography method.