Vandermonde Wave Function Ansatz for Improved Variational Monte Carlo

Abstract

Solutions to the Schrödinger equation can be used to predict the electronic structure of molecules and materials and therefore infer their complex physical and chemical properties. Variational Quantum Monte Carlo (VMC) is a technique that can be used to solve the weak form of the Schrödinger equation. Applying VMC to systems with N electrons involves evaluating the determinant of an N by N matrix. The evaluation of this determinant scales as $O(N^{3})$ and is the main computational cost in the VMC process. In this work, we investigate an alternative VMC technique based on the Vandermonde determinant. The Vandermonde determinant is a product of pairwise differences and so evaluating it scales as $O(N^{2})$. Therefore, this approach reduces the computational cost by a factor of N. The Vandermonde determinant was implemented in PyTorch and the performance was assessed in approximating the ground state energy of various quantum systems against existing techniques. The performance is evaluated in a variety of systems, starting with the one-dimensional particle in a box, and then considering more complicated atomic systems with multiple particles. The Vandermonde determinant was also implemented in PauliNet, a deep-learning architecture for VMC. The new method is shown to be computationally efficient, and results in a speed-up as large as 5X. In these cases, the new ansatz obtains a reasonable approximation for wavefunctions of atomic systems, but does not reach the accuracy of the Hartree-Fock method that relies on the Slater determinant. It is observed that while the use of neural networks in VMC can result in highly accurate solutions, further work is necessary to determine an appropriate balance between computational time and accuracy.