  Motivated by recent experiments with a Penning ion trap quantum simulator, we
perform numerically exact Stochastic Series Expansion quantum Monte Carlo
simulations of long-range transverse-field Ising models on a triangular lattice
for different decay powers $\alpha$ of the interactions. The phase boundary for
the ferromagnet is obtained as a function of $\alpha$. For antiferromagnetic
interactions, there is strong indication that the transverse field stabilizes a
clock ordered phase with sublattice magnetization $(M,-\frac{M}{2},
-\frac{M}{2})$ with unsaturated $M < 1$ in a process known as "order by
disorder" similar to the nearest neighbour antiferromagnet on the triangular
lattice. Connecting the known limiting cases of nearest neighbour and
infinite-range interactions, a semiquantitative phase diagram is obtained.
Magnetization curves for the ferromagnet for experimentally relevant system
sizes and with open boundary conditions are presented.
