  Variational quantum eigensolver (VQE) is an efficient classical-quantum
hybrid method to take advantage of quantum computers in the Noisy
Intermediate-Scale Quantum (NISQ) era. In this work we test the performance of
VQE by studying the $J_1$-$J_2$ anti-ferromagnetic Heisenberg model on the
kagome lattice, which is found to display a first order phase transition at
$J_2 / J_1 \approx 0.01$. By comparing the VQE states with the exact
diagonalization results, we find VQE energies agree well with the exact values
in most region of parameters for the 18-site system we studied. However, near
the phase transition point, VQE tends to converge to the excited states when
the number of variational parameters is not large enough. For the system
studied in this work, this issue can be solved by either increasing the number
of parameters or by initializing the parameters with converged values for
$J_2/J_1$ away from the phase transition point. Our results provide useful
guidance for the practical application of VQE on real quantum computers to
study strongly correlated quantum many-body systems.
