Determinant-free fermionic wave function using feed-forward neural networks

In recent years, there has been a remarkable increase in research on approximating many-body wave functions by neural networks. It is, however, still challenging to find the ground state of fermionic many-body systems due to the complex sign structure arising from the anticommutation relation between fermions. The sign structure is usually implemented to a variational wave function by the Slater determinant (or Pfaffian), which is a computational bottleneck because of the numerical cost of O(N³) for N particles. We here propose a framework to bypass this bottleneck by explicitly calculating the sign changes associated with particle exchanges in real space and using fully connected neural networks for optimizing the rest parts of the wave function. This reduces the computational cost to O(N²) or less. In addition, we device some numerical tricks for stabilization of the calculations, e.g., a reweighting method in Monte Carlo sampling. We apply our method to the Hubbard model with 6x6 lattice sites and N=10, and find that it achieves a lower energy than the many-variable variational Monte Carlo calculation.