Prospects for Quantum Equivariant Neural Networks

Abstract

Convolutional neural networks (CNNs) exploit translational invariance within images. Group equivariant neural networks comprise a natural generalization of convolutional neural networks by exploiting other symmetries arising through different group actions. Informally, a linear map is equivariant if it transfers symmetries from its input space into its output space. Equivariant neural networks guarantee equivariance for arbitrary groups, reducing the system design complexity. Motivated by the theoretical/experimental development of quantum computing, in particular with the quantum advantage derived from other quantum algorithms/subroutines for group theoretic and linear algebraic problems, we explore the potential of quantum computers to realize these structures in machine learning. This work reviews the mathematical machinery necessary from group representation theory, surveys the theory of equivariance, and combines results in non-commutative harmonic analysis and geometric deep learning. Convolutions and cross-correlations are examples of functions which are equivariant to the actions of a group. We present efficient quantum algorithms for performing linear finite-group convolutions and cross-correlations on data stored as quantum states. Potential implementations and quantizations of the infinite group cases also discussed.