Solving the Hubbard model with neural quantum states

The rapid development of neural quantum states (NQS) has established it as a promising framework for studying quantum many-body systems. In this work, by leveraging the transformer-based architectures and developing efficient optimization algorithms, we achieve the state-of-the-art results for the doped two-dimensional (2D) Hubbard model, arguably the minimum model for high-Tc superconductivity. Interestingly, we find different attention heads in the NQS ansatz can directly encode correlations at different scales, making it capable of capturing long-range correlations in strongly correlated systems. With these advances, we find evidence for the half-filled stripe in the ground state of 2D Hubbard model with the next nearest neighboring hopping, consistent with experimental observations in cuprates. Our work establishes NQS as a powerful tool for solving challenging many-fermions systems.