  We study the effect of quantum fluctuations by means of a transverse magnetic
field ($\Gamma$) on the antiferromagnetic $J_1-J_2$ Ising model on the
checkerboard lattice, the two dimensional version of the pyrochlore lattice.
The zero-temperature phase diagram of the model has been obtained by employing
a plaquette operator approach (POA). The plaquette operator formalism bosonizes
the model, in which a single boson is associated to each eigenstate of a
plaquette and the inter-plaquette interactions define an effective Hamiltonian.
The excitations of a plaquette would represent an-harmonic fluctuations of the
model, which lead not only to lower the excitation energy compared with a
single-spin flip but also to lift the extensive degeneracy in favor of a
plaquette ordered solid (RPS) state, which breaks lattice translational
symmetry, in addition to a unique collinear phase for $J_2>J_1$. The bosonic
excitation gap vanishes at the critical points to the N\'{e}el ($J_2 < J_1$)
and collinear ($J_2 > J_1$) ordered phases, which defines the critical phase
boundaries. At the homogeneous coupling ($J_2=J_1$) and its close neighborhood,
the (canted) RPS state, established from an-harmonic fluctuations, lasts for
low fields, $\Gamma/J_1\lesssim 0.3$, which is followed by a transition to the
quantum paramagnet (polarized) phase at high fields. The transition from RPS
state to the N\'{e}el phase is either a deconfined quantum phase transition or
a first order one, however a continuous transition occurs between RPS and
collinear phases.
