  We present a Quantum Monte Carlo study of the Ising model in a transverse
field on a square lattice with nearest-neighbor antiferromagnetic exchange
interaction J and one diagonal second-neighbor interaction $J'$, interpolating
between square-lattice ($J'=0$) and triangular-lattice ($J'=J$) limits. At a
transverse-field of $B_x=J$, the disorder-line first introduced by Stephenson,
where the correlations go from Neel to incommensurate, meets the zero
temperature axis at $J'\approx 0.7 J$. Strong evidence is provided that the
incommensurate phase at larger $J'$, at finite temperatures, is a floating
phase with power-law decaying correlations. We sketch a general phase-diagram
for such a system and discuss how our work connects with the previous Quantum
Monte Carlo work by Isakov and Moessner for the isotropic triangular lattice
($J'=J$). For the isotropic triangular-lattice, we also obtain the entropy
function and constant entropy contours using a mix of Quantum Monte Carlo,
high-temperature series expansions and high-field expansion methods and show
that phase transitions in the model in presence of a transverse field occur at
very low entropy.
