  Efficient discretisations of gauge groups are crucial with the long term
perspective of using tensor networks or quantum computers for lattice gauge
theory simulations. For any Lie group other than U$(1)$, however, there is no
class of asymptotically dense discrete subgroups. Therefore, discretisations
limited to subgroups are bound to lead to a freezing of Monte Carlo simulations
at weak couplings, necessitating alternative partitionings without a group
structure. In this work we provide a comprehensive analysis of this freezing
for all discrete subgroups of SU$(2)$ and different classes of asymptotically
dense subsets. We find that an appropriate choice of the subset allows unfrozen
simulations for arbitrary couplings, though one has to be careful with varying
weights of unevenly distributed points. A generalised version of the Fibonacci
spiral appears to be particularly efficient and close to optimal.
