Cloning is as Hard as Learning for Stabilizer States

Cloning non-orthogonal states is fundamentally limited by the principles of quantum theory, and even approximate cloning of arbitrary unknown pure states demands a substantial number of initial copies, equivalent to the amount required for complete state learning. Recent research in quantum learning theory has focused on structured state classes, leveraging their inherent properties to improve learning outcomes. However, the core challenge of cloning remains, with the number of required initial copies directly tied to the learning process. This finding underscores the intrinsic connection between cloning and learning in quantum systems, highlighting the stringent requirements for approximating unknown states¹. The implications of this research are significant, as they impose strict limitations on the capabilities of quantum cloning protocols, making it essential for practitioners to carefully consider the trade-offs between cloning fidelity and resource requirements. This fundamental constraint has far-reaching consequences for quantum information processing and quantum computing applications.