  Variational approaches are among the most powerful modern techniques to
approximately solve quantum many-body problems. These encompass both
variational states based on tensor or neural networks, and parameterized
quantum circuits in variational quantum eigensolvers. However, self-consistent
evaluation of the quality of variational wavefunctions is a notoriously hard
task. Using a recently developed Hamiltonian reconstruction method, we propose
a multi-faceted approach to evaluating the quality of neural-network based
wavefunctions. Specifically, we consider convolutional neural network (CNN) and
restricted Boltzmann machine (RBM) states trained on a square lattice spin-1/2
$J_1$-$J_2$ Heisenberg model. We find that the reconstructed Hamiltonians are
typically less frustrated, and have easy-axis anisotropy near the high
frustration point. Furthermore, the reconstructed Hamiltonians suppress quantum
fluctuations in the large $J_2$ limit. Our results highlight the critical
importance of the wavefunction's symmetry. Moreover, the multi-faceted insight
from the Hamiltonian reconstruction reveals that a variational wave function
can fail to capture the true ground state through suppression of quantum
fluctuations.
