Multi-copy prediction in quantum machine learning

Quantum machine learning (QML) promises advantages in pattern recognition over classical models by learning on quantum data. However, its performance in realistic settings, where only a finite number of copies of quantum data are available, remains poorly understood. Here we address the problem that bridges classical machine learning, where classical instances - effectively infinite number of copies - are available, and quantum hypothesis testing, which concerns the discrimination of density matrices of single-copy quantum states. Specifically, we study optimal strategies for identifying a data class by measuring a limited number of copies of a single instance in the test dataset. We evaluate the quantum limits on the prediction error of two-class and multi-class classification tasks with a given positive operator-valued measure (POVM) and develop an efficient algorithm to optimize the measurement that minimizes empirical risk on benchmark datasets. We constrain the POVM to be either separable (independent and identically distributed single-copy) or few-copy entangled measurements, aligning with experimentally accessible architectures. Our results show that both the prediction error and the optimal classifier depend sensitively on the dataset, system dimension, and the number of copies N. In the multi-copy setting, the optimal classifier differs qualitatively from the single-copy optimum, revealing a clear quantum-to-classical transition in the machine learning task. Our result paves the way for assessing realistic performance and advantage of QML models and designing practical QML protocols.