Scalable and accurate simulations of the Hubbard model with neural quantum states

Neural quantum states (NQS) recently had a major impact on quantum many-body physics by suggesting novel wave-functions parametrized by neural networks. Nevertheless, NQS faces great challenges in fermionic systems, especially the Fermi-Hubbard model (FHM), a minimal model for unconventional superconductivity (SC). FHM poses three major numerical difficulties: large system sizes, correlated ground states, and competing low-energy states. In this work, we introduce the hidden fermion Pfaffian state (HFPS), which integrates ANN with Pfaffian that naturally encodes SC pairings. HFPS alleviates the three problems and provides state-of-the-art accuracy in FHM. Our results highlight the effectiveness of NQS and point toward promising future directions for resolving the longstanding challenge of unconventional SC in FHM.