  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.
