Accurate simulations of the Hubbard model are crucial to understanding strongly correlated phenomena, in which small energy differences between competing orders demand high numerical precision. In this work, neural quantum states are used to probe the strongly coupled and underdoped regime of the square-lattice Hubbard model. We systematically compare the hidden fermion determinant state and the Jastrow-Backflow Ansatz, parametrized by a vision transformer, finding that in practice, their accuracies are similar. We also test different symmetrization strategies, finding that output averaging yields the lowest energies, although it becomes costly for larger system sizes. On cylindrical systems, we consistently observe filled stripes. On the torus, our calculations display features consistent with a doped Mott insulator, including antiferromagnetic correlations and suppressed density fluctuations. Our results demonstrate both the promise and current challenges of neural quantum states for correlated fermions.

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• Strongly correlated systems
• Hubbard model
• Neural network simulations
• Variational wave functional methods