Neural Quantum States: Fermions on D-Dimensions

Abstract

In this work, we explore the use of Neural Quantum States to approximate the ground-state wavefunctions of fully polarized fermionic systems confined in a D-dimensional harmonic trap. Building on the architecture introduced in [1], we generalize the input representation and network structure to handle arbitrary spatial dimensionality, extending the applicability of the method beyond one-dimensional systems. The antisymmetric nature of the fermionic wavefunction is preserved through the use of equivariant neural layers, and a generalized Slater determinant is constructed from learned single-particle orbitals modulated by a Gaussian envelope. Training is carried out in two stages: first, a supervised pretraining phase based on analytical solutions of the non-interacting system, which is then followed by variational Monte Carlo optimization of the network parameters using the energy as the loss function. We validate our approach on non-interacting systems with up to 4 particles in 2D and 3 particles in 3D, where analytical solutions are available for benchmarking. Results show excellent agreement in terms of mean energy, one-body density, and the one-body density matrix, with observed spatial symmetries and degeneracy patterns matching theoretical expectations. While the training protocol has been generalized to incorporate finite-range interactions, this study focuses on non-interacting systems to establish a solid baseline. The framework developed here provides a flexible and scalable foundation for future exploration of interacting quantum systems in higher dimensions using neural variational methods.