  We investigate the ground-state phase diagram of the frustrated transverse
field Ising (TFI) model on the checkerboard lattice (CL), which consists of
N\'{e}el, collinear, quantum paramagnet and plaquette-valence bond solid (VBS)
phases. We implement a numerical simulation that is based on the recently
developed unconstrained tree tensor network (TTN) ansatz, which systematically
improves the accuracy over the conventional methods as it exploits the internal
gauge selections. At the highly frustrated region ($J_2=J_1$), we observe a
second order phase transition from plaquette-VBS state to paramagnet phase at
the critical magnetic field, $\Gamma_{c}=0.28$, with the associated critical
exponents $\nu=1$ and $\gamma\simeq0.4$, which are obtained within the finite
size scaling analysis on different lattice sizes $N=4\times 4, 6\times 6,
8\times8$. The stability of plaquette-VBS phase at low magnetic fields is
examined by spin-spin correlation function, which verifies the presence of
plaquette-VBS at $J_2=J_1$ and rules out the existence of a N\'{e}el phase. In
addition, our numerical results suggest that the transition from N\'{e}el (for
$J_2<J_1$) to plaquette-VBS phase is a deconfined phase transition. Moreover,
we introduce a mapping, which renders the low-energy effective theory of TFI on
CL to be the same model on $J_1-J_2$ square lattice (SL). We show that the
plaquette-VBS phase of the highly frustrated point $J_2=J_1$ on CL is mapped to
the emergent string-VBS phase on SL at $J_2=0.5J_1$.
