Quantum algorithms for group convolution, cross-correlation, and equivariant transformations

TL;DR

This paper introduces quantum algorithms for group convolutions and cross-correlations that are exponentially faster than classical methods, enabling efficient processing of symmetric data in quantum machine learning.

Contribution

It provides the first efficient quantum algorithms for linear group convolutions and cross-correlations with logarithmic runtime dependence on group size.

Findings

Algorithms achieve exponential speedup over classical counterparts.

Runtime is logarithmic in the size of the group.

Framework enables quantum acceleration of symmetry-based algorithms.

Abstract

Group convolutions and cross-correlations, which are equivariant to the actions of group elements, are commonly used in mathematics to analyze or take advantage of symmetries inherent in a given problem setting. Here, we provide efficient quantum algorithms for performing linear group convolutions and cross-correlations on data stored as quantum states. Runtimes for our algorithms are logarithmic in the dimension of the group thus offering an exponential speedup compared to classical algorithms when input data is provided as a quantum state and linear operations are well conditioned. Motivated by the rich literature on quantum algorithms for solving algebraic problems, our theoretical framework opens a path for quantizing many algorithms in machine learning and numerical methods that employ group operations.