Advanced Micro Devices (AMD) Inc Austin, United States of America
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 x N matrix. The evaluation of this determinant scales as 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 N^2. Therefore, our approach reduces the computational cost by a factor of N.
We implemented VMC using the new low cost approach in PyTorch and compared its use in approximating the ground state energy of various quantum systems against existing techniques, starting with the one-dimensional particle in a box and moving on to more complicated atomic systems with multiple particles. We also implemented the Vandermonde determinant as a part of PauliNet, a deep-learning architecture for VMC. While the new method is computationally efficient and obtains a reasonable approximation for wavefunctions of atomic systems, it does not reach the accuracy of the Hartree-Fock method that relies on the Slater determinant. We observed that while the use of neural networks in VMC can result in highly accurate solutions, further new approaches are needed to best balance computational cost with accuracy.