Group-Invariant Quantum Machine Learning

TL;DR

This paper introduces a framework for quantum machine learning models that incorporate problem symmetries to improve trainability and generalization, demonstrating theoretical foundations and practical applications across various symmetry groups.

Contribution

It presents a novel group-invariant framework for QML models that respects data symmetries, unifying and extending existing algorithms with a geometric approach.

Findings

Framework respects data symmetries in QML models

Applicable to continuous Lie and discrete groups

Recovers and extends known algorithms

Abstract

Quantum Machine Learning (QML) models are aimed at learning from data encoded in quantum states. Recently, it has been shown that models with little to no inductive biases (i.e., with no assumptions about the problem embedded in the model) are likely to have trainability and generalization issues, especially for large problem sizes. As such, it is fundamental to develop schemes that encode as much information as available about the problem at hand. In this work we present a simple, yet powerful, framework where the underlying invariances in the data are used to build QML models that, by construction, respect those symmetries. These so-called group-invariant models produce outputs that remain invariant under the action of any element of the symmetry group $\mathfrak{G}$ associated to the dataset. We present theoretical results underpinning the design of $\mathfrak{G}$-invariant models, and exemplify their application through several paradigmatic QML classification tasks including cases when $\mathfrak{G}$ is a continuous Lie group and also when it is a discrete symmetry group. Notably, our framework allows us to recover, in an elegant way, several well known algorithms for the literature, as well as to discover new ones. Taken together, we expect that our results will help pave the way towards a more geometric and group-theoretic approach to QML model design.