Generalization Bounds for Quantum Learning via R\'enyi Divergences

TL;DR

This paper develops new theoretical bounds on the generalization error of quantum learning algorithms using quantum and classical Re9nyi divergences, with improved bounds demonstrated through analysis and numerics.

Contribution

It introduces a novel family of generalization bounds for quantum learning based on Re9nyi divergences, including a new modified sandwich divergence, advancing theoretical understanding.

Findings

Bounds based on the modified sandwich quantum Re9nyi divergence outperform those based on Petz divergence.

Analytical and numerical results demonstrate the effectiveness of the new bounds.

Two probabilistic generalization error bounds are established using different divergence techniques.

Abstract

This work advances the theoretical understanding of quantum learning by establishing a new family of upper bounds on the expected generalization error of quantum learning algorithms, leveraging the framework introduced by Caro et al. (2024) and a new definition for the expected true loss. Our primary contribution is the derivation of these bounds in terms of quantum and classical R\'enyi divergences, utilizing a variational approach for evaluating quantum R\'enyi divergences, specifically the Petz and a newly introduced modified sandwich quantum R\'enyi divergence. Analytically and numerically, we demonstrate the superior performance of the bounds derived using the modified sandwich quantum R\'enyi divergence compared to those based on the Petz divergence. Furthermore, we provide probabilistic generalization error bounds using two distinct techniques: one based on the modified sandwich quantum R\'enyi divergence and classical R\'enyi divergence, and another employing smooth max R\'enyi divergence.