We investigate the application of efficient recursive numerical integration strategies to models in lattice gauge theory from quantum field theory. Given the coupling structure of the physics problems and the group structure within lattice cubature rules for numerical integration, we show how to approach these problems efficiently by means of Fast Fourier Transform techniques. In particular, we consider applications to the quantum mechanical rotor and compact $U(1)$ lattice gauge theory, where the physical dimensions are two and three. This proceedings article reviews our results presented in J. Comput. Phys 443 (2021) 110527.