  We examine applicability of the valence bond basis correlator product state
ansatz, equivalent to the restricted Boltzmann machine quantum artificial
neural network ansatz, and variational Monte Carlo method for direct
optimization of excited energy states to study properties of strongly
correlated and frustrated quantum systems. The energy eigenstates are found by
stochastic minimization of the variational function for the energy eigenstates
which allows direct optimization of particular energy state without knowledge
of the lower energy states. This approach combined with numerous tensor network
or artificial neural network ansatz wavefunctions then allows further insight
into quantum phases and phase transitions in various strongly correlated models
by considering properties of these systems beyond the ground state properties.
Also, the method is in general applicable to any dimension and has no sign
instability. An example that we consider is the square lattice J1-J2
antiferromagnetic Heisenberg model. The model is one of the most studied models
in frustrated quantum magnetism since it is closely related to the
disappearance of the antiferromagnetic order in the high-Tc superconducting
materials and there is still no agreement about the properties of the system in
the highly frustrated regime near J2/J1 = 0.5. For J1-J2 model we write the
variational ansatz in terms of the two site correlators and in the valence bond
basis and calculate lowest energy eigenstates in the highly frustrated regime
near J2/J1 = 0.5 where the system has a paramagnetic phase. We find that our
results are in good agreement with previously obtained results which confirms
applicability of the method to study frustrated spin systems.
