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We examine applicability of the valence bond basis correlator product stateansatz, equivalent to the restricted Boltzmann machine quantum artificialneural network ansatz, and variational Monte Carlo method for directoptimization of excited energy states to study properties of stronglycorrelated and frustrated quantum systems. The energy eigenstates are found bystochastic minimization of the variational function for the energy eigenstateswhich allows direct optimization of particular energy state without knowledgeof the lower energy states. This approach combined with numerous tensor networkor artificial neural network ansatz wavefunctions then allows further insightinto quantum phases and phase transitions in various strongly correlated modelsby considering properties of these systems beyond the ground state properties.Also, the method is in general applicable to any dimension and has no signinstability. An example that we consider is the square lattice J1-J2antiferromagnetic Heisenberg model. The model is one of the most studied modelsin frustrated quantum magnetism since it is closely related to thedisappearance of the antiferromagnetic order in the high-Tc superconductingmaterials and there is still no agreement about the properties of the system inthe highly frustrated regime near J2/J1 = 0.5. For J1-J2 model we write thevariational ansatz in terms of the two site correlators and in the valence bondbasis and calculate lowest energy eigenstates in the highly frustrated regimenear J2/J1 = 0.5 where the system has a paramagnetic phase. We find that ourresults are in good agreement with previously obtained results which confirmsapplicability of the method to study frustrated spin systems.

Efficient neural-network based variational Monte Carlo scheme for direct optimization of excited energy states in frustrated quantum systems