Digitization and subduction of $SU(N)$ gauge theories

The simulation of lattice gauge theories on quantum computers necessitates digitizing gauge fields. One approach involves substituting the continuous gauge group with a discrete subgroup, but the implications of this approximation still need to be clarified. To gain insights, we investigate the subduction of $SU(2)$ and $SU(3)$ to discrete crystal-like subgroups. Using classical lattice calculations, we show that subduction offers valuable information based on subduced direct sums, helping us identify additional terms to incorporate into the lattice action that can mitigate the effects of digitization. Furthermore, we compute the static potentials of all irreducible representations of $\Sigma(360 \times 3)$ at a fixed lattice spacing. Our results reveal a percent-level agreement with the Casimir scaling of ( SU(3) ) for irreducible representations that subduce to a single $\Sigma(360 \times 3)$ irreducible representation. This provides a diagnostic measure of approximation quality, as some irreducible representations closely match the expected results while others exhibit significant deviations.