Representation Theory for Geometric Quantum Machine Learning

TL;DR

This paper introduces the use of representation theory to incorporate symmetries into quantum machine learning models, aiming to enhance their performance and problem-specific capabilities.

Contribution

It provides an accessible overview of representation theory tools and methods for implementing symmetry-aware architectures in Geometric Quantum Machine Learning.

Findings

Illustrates how group representation theory underpins GQML architectures

Demonstrates the use of Haar integration and twirling in symmetry detection

Provides practical strategies for embedding symmetries in quantum models

Abstract

Recent advances in classical machine learning have shown that creating models with inductive biases encoding the symmetries of a problem can greatly improve performance. Importation of these ideas, combined with an existing rich body of work at the nexus of quantum theory and symmetry, has given rise to the field of Geometric Quantum Machine Learning (GQML). Following the success of its classical counterpart, it is reasonable to expect that GQML will play a crucial role in developing problem-specific and quantum-aware models capable of achieving a computational advantage. Despite the simplicity of the main idea of GQML -- create architectures respecting the symmetries of the data -- its practical implementation requires a significant amount of knowledge of group representation theory. We present an introduction to representation theory tools from the optics of quantum learning, driven by key examples involving discrete and continuous groups. These examples are sewn together by an exposition outlining the formal capture of GQML symmetries via "label invariance under the action of a group representation", a brief (but rigorous) tour through finite and compact Lie group representation theory, a reexamination of ubiquitous tools like Haar integration and twirling, and an overview of some successful strategies for detecting symmetries.