# Information-theoretic generalization bounds for learning from quantum data

**Authors:** Matthias Caro, Tom Gur, Cambyse Rouzé, Daniel Stilck França, Sathyawageeswar Subramanian  
**arXiv:** [2311.05529](https://arxiv.org/abs/2311.05529)  
**Date:** 2023-11-09  
**Summary page:** https://www.summarizepaper.com/en/arxiv-id/2311.05529v1/

## Key points

- Authors Matthias Caro, Tom Gur, Cambyse Rouzé, Daniel Stilck França, and Sathyawageeswar Subramanian explore the role of learning tasks in quantum information and computation.
- Fundamental issues highlighted include state discrimination, metrology, quantum PAC learning, and shadow variants of state tomography.
- Proposal of a mathematical formalism to describe quantum learning through training on classical-quantum data and testing generalization capabilities on new data.
- Establishment of bounds on expected generalization error by relating it to classical and quantum information-theoretic quantities.
- Utilization of tools from quantum optimal transport and quantum concentration inequalities to create non-commutative versions of decoupling lemmas.
- Provision of accessible generalization bounds for various quantum learning scenarios such as state discrimination, PAC learning of quantum states, parameter estimation, and PAC learning of classical functions with a quantum twist.
- Contribution towards unifying perspectives on quantum learning from an information-theoretic standpoint and bridging gaps between different branches of quantum learning theory.

## Summary

In their paper titled "Information-theoretic generalization bounds for learning from quantum data," authors Matthias Caro, Tom Gur, Cambyse Rouzé, Daniel Stilck França, and Sathyawageeswar Subramanian delve into the expanding role of learning tasks in quantum information and computation. They highlight a variety of fundamental issues within this realm, such as state discrimination and metrology, as well as the framework of quantum probably approximately correct (PAC) learning and shadow variants of state tomography. The authors acknowledge that the field of quantum learning theory has developed along various separate paths thus far. To address this fragmentation, they propose a comprehensive mathematical formalism that describes quantum learning through training on classical-quantum data and subsequently testing the generalization capabilities of the acquired hypothesis on new data. By establishing bounds on the expected generalization error of a quantum learner in relation to classical and quantum information-theoretic quantities, they aim to quantify how strongly the learner's hypothesis relies on specific training data. To achieve their goals, the authors employ tools from quantum optimal transport and quantum concentration inequalities to create non-commutative versions of decoupling lemmas. These lemmas serve as foundational elements for recent information-theoretic generalization bounds in classical machine learning. Through their framework, they provide accessible generalization bounds for a range of quantum learning scenarios including but not limited to quantum state discrimination, PAC learning of quantum states, quantum parameter estimation, and PAC learning of classical functions with a quantum twist. Overall,this work by Caro et al.lays down a solid foundation for a unified perspective on quantum learning from an information-theoretic standpoint.Their efforts contribute significantly to bridging the gaps between different branches of quantum learning theory and pave the way for further advancements in this rapidly evolving field.