Equivalence of Classical and Sampling Advice in Learning Theory

In a recent paper titled "On classical advice, sampling advise and complexity assumptions for learning separations," author Jordi Pérez-Guijarro explores the relationship between classical advice and sampling advice in computational learning theory. The paper, submitted on August 25, 2024, presents a significant finding: it establishes the equivalence between sampling advice, which is essentially advice in the form of a training set, and classical advice. The main result indicates that BPP/samp is equal to P/poly, a notable conclusion in the field of computational complexity.

The study further investigates these relationships under the constraint of a fixed distribution, revealing that the equality does not hold in such cases. This finding is also applicable when considering quantum advice and a quantum generalization of the training set. The author identifies necessary and sufficient complexity assumptions for the existence of concept classes that can exhibit a quantum learning speed-up in worst-case scenarios, where accurate results are required for all inputs.

These insights could have implications for the development of more efficient learning algorithms in quantum computing, potentially enhancing the capabilities of quantum systems in processing and learning from data. The paper contributes to the ongoing discourse on the intersection of classical and quantum computational theories, providing a framework for future research in this area.

For further details, the paper can be accessed at arXiv:2408.13880.