  Efficient digitization is required for quantum simulations of gauge theories.
Schemes based on discrete subgroups use fewer qubits at the cost of systematic
errors. We systematize this approach by deriving a single plaquette action for
approximating general continuous gauge groups through integrating out field
fluctuations. This provides insight into the effectiveness of these
approximations, and how they could be improved. We accompany the scheme by
simulations of pure gauge over the largest discrete subgroup of $SU(3)$ up to
the third order.
