Separable Power of Classical and Quantum Learning Protocols Through the Lens of No-Free-Lunch Theorem

TL;DR

This paper extends the No-Free-Lunch theorem to quantum machine learning, categorizing protocols and demonstrating quadratic sample complexity reductions, revealing quantum advantages in learning quantum dynamics.

Contribution

It establishes the NFL theorem for quantum learning protocols and compares classical, restricted quantum, and full quantum protocols, highlighting quantum advantages.

Findings

Quadratic reductions in sample complexity for quantum protocols

Quantum protocols leverage global phase information

Deeper understanding of quantum vs classical learning capabilities

Abstract

The No-Free-Lunch (NFL) theorem, which quantifies problem- and data-independent generalization errors regardless of the optimization process, provides a foundational framework for comprehending diverse learning protocols' potential. Despite its significance, the establishment of the NFL theorem for quantum machine learning models remains largely unexplored, thereby overlooking broader insights into the fundamental relationship between quantum and classical learning protocols. To address this gap, we categorize a diverse array of quantum learning algorithms into three learning protocols designed for learning quantum dynamics under a specified observable and establish their NFL theorem. The exploited protocols, namely Classical Learning Protocols (CLC-LPs), Restricted Quantum Learning Protocols (ReQu-LPs), and Quantum Learning Protocols (Qu-LPs), offer varying levels of access to quantum resources. Our derived NFL theorems demonstrate quadratic reductions in sample complexity across CLC-LPs, ReQu-LPs, and Qu-LPs, contingent upon the orthogonality of quantum states and the diagonality of observables. We attribute this performance discrepancy to the unique capacity of quantum-related learning protocols to indirectly utilize information concerning the global phases of non-orthogonal quantum states, a distinctive physical feature inherent in quantum mechanics. Our findings not only deepen our understanding of quantum learning protocols' capabilities but also provide practical insights for the development of advanced quantum learning algorithms.