The Performance of VQE across a phase transition point in the $J_1$-$J_2$ model on kagome lattice

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.