Machine Learning as an Alternative Wavefunction Ansatz to Improve Variational Monte Carlo
Solutions to the Schrodinger equation can be used to predict the electronic structure of molecules and materials and therefore infer their complex physical and chemical properties. One area of application is the design of semiconductors for computer chips. Variational Quantum Monte Carlo (VMC) is a technique that can be used to solve the weak form of the Schrodinger 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(N3) and is the main computational cost in the VMC process. In this work we propose 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(N2). 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 use the Vandermonde determinant as a part of PauliNet, a deep-learning architecture for VMC. While the new method obtains a reasonable approximation for wavefunctions of atomic systems, it does not reach the accuracy of the Hartree-Fock method that relies on the standard 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.