New Bounds on Quantum Sample Complexity of Measurement Classes
Mohsen Heidari, Wojciech Szpankowski
TL;DR
The paper addresses the problem of quantum supervised learning for classical inference from quantum states under the QPAC framework, where quantum sample complexity is challenged by no-cloning, state collapse, and measurement incompatibility. It introduces a shadow tomography–based quantum ERM (QSRM) that focuses on extreme points $\mathcal{C}^*$ of the convex closure of a concept class $\mathcal{C}$ and leverages the shadow-norm to bound sample complexity. The main result establishes a bound $n_{\mathcal{C}}(\epsilon,\delta) = O\bigl(V_{\mathcal{C}^*} / \epsilon^2 \cdot \log(|\mathcal{C}^*|/\delta)\bigr)$, with tightness in bounded-norm settings reducing to $O(\log |\mathcal{C}^*|)$, and provides a scalable method that circumvents full sample duplication. This advances quantum learning by reducing dependence on the full class size and offering a practical framework for learning measurement classes under quantum constraints.
Abstract
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 $O(|\mathcal{C}|)$. This is significantly higher than the classical sample complexity that grows logarithmically with the class size. This work improves the sample complexity bound to $O(V_{\mathcal{C}^*} \log |\mathcal{C}^*|)$, 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 $O(\log |\mathcal{C}^*|)$. Our approach is based on a new quantum empirical risk minimization (ERM) algorithm equipped with a shadow tomography method.