  We propose an efficient numerical method, which combines the advantages of
recently developed tensor-network based methods and standard trial wave
functions, to study the ground state properties of quantum many-body systems.
In this approach, we apply a projector in the form of a tensor-product operator
to an input wave function, such as a Jastrow-type or Hartree-Fock wave
function, and optimize the tensor elements via variational Monte Carlo. The
entanglement already contained in the input wave function can considerably
reduce the bond dimensions compared to the regular tensor-product state
representation. In particular, this allows us to also represent states that do
not obey the area law of entanglement entropy. In addition, for fermionic
systems, the fermion sign structure can be encoded in the input wave function.
We show that the optimized states provide good approximations of the
ground-state energy and correlation functions in the cases of two-dimensional
bosonic and fermonic systems.
