  We investigate the ground state nature of the transverse field Ising model on
the $J_1-J_2$ square lattice at the highly frustrated point $J_2/J_1=0.5$. At
zero field, the model has an exponentially large degenerate classical ground
state, which can be affected by quantum fluctuations for non-zero field toward
a unique quantum ground state. We consider two types of quantum fluctuations,
harmonic ones by using linear spin wave theory (LSWT) with single-spin flip
excitations above a long range magnetically ordered background and anharmonic
fluctuations, by employing a cluster-operator approach (COA) with multi-spin
cluster type fluctuations above a non-magnetic cluster ordered background. Our
findings reveal that the harmonic fluctuations of LSWT fail to lift the
extensive degeneracy as well as signaling a violation of the Hellmann-Feynman
theorem. However, the string-type anharmonic fluctuations of COA are able to
lift the degeneracy toward a string-valence bond solid (VBS) state, which is
obtained from an effective theory consistent with the Hellmann-Feynman theorem
as well. Our results are further confirmed by implementing numerical tree
tensor network simulation. The emergent non-magnetic string-VBS phase is gapped
and breaks lattice rotational symmetry with only two-fold degeneracy, which
bears a continuous quantum phase transition at $\Gamma/J_1 \cong 0.50$ to the
quantum paramagnet phase of high fields. The critical behavior is characterized
by $\nu \cong 1.0$ and $\gamma \cong 0.33$ exponents.
