  The tensor network algorithm, a family of prevalent numerical methods for
quantum many-body problems, aptly captures the entanglement properties
intrinsic to quantum systems, enabling precise representation of quantum
states. However, its computational cost is notably high, particularly in
calculating physical observables like correlation functions. To surmount the
computational challenge and enhance efficiency, we propose integrating the
Green's function Monte Carlo (GFMC) method with the projected entangled pair
state (PEPS) ansatz. This approach combines the high-efficiency characteristics
of Monte Carlo with the sign-free nature of tensor network states and proves
effective in addressing the computational bottleneck. To showcase its prowess,
we apply this hybrid approach to investigate the antiferromagnetic $J_1$-$J_2$
Heisenberg model on the square lattice, a model notorious for its sign problem
in quantum Monte Carlo simulations. Our results reveal a substantial
improvement in the accuracy of ground-state energy when utilizing a preliminary
PEPS as the guiding wave function for GFMC. By calculating the structure factor
and spin-spin correlation functions, we further characterize the phase diagram,
identifying a possible columnar valence-bond state phase within the
intermediate parameter range of $0.52 < J_2/J_1 < 0.58$. This comprehensive
study underscores the efficacy of our combined approach, demonstrating its
ability to accurately simulate frustrated quantum spin systems while ensuring
computational efficiency.
