  We study the low-field ground-state (GS) properties of the antiferromagnetic
transverse-field Ising model with long-range interactions (afLRTFIM) on the
triangular lattice. We use the method of perturbative continuous unitary
transformations (pCUT) to derive an effective model for the degenerate GS space
of the antiferromagnetic nearest-neighbour (NN) Ising model on a finite system,
by treating the transverse-field (TF) and the long-range interactions (LRI) as
a perturbation. We determine a level-crossing between the plain stripe phase at
small TF and the clock-ordered phase at intermediate TF at $h\cong0.129$ for
$\alpha=6$, $N=36$ spins in order three perturbation theory. We discuss the
qualitative layout of the quantum phase diagram of the afLRTFIM on the
triangular lattice.
