Pareto Frontier of Neural Quantum States: Scalable, Affordable, and Accurate Convolutional Backflow for Strongly Correlated Lattice Fermions

The recent development of Neural Quantum States (NQS) has established them as one of the most accurate methods for studying strongly correlated many-fermion systems, outperforming existing many-body approaches for large systems. However, NQS calculations are currently highly resource intensive. In this work, we introduce an improved Pareto frontier of efficiency and accuracy for NQS in the simulation of strongly correlated lattice fermions. This frontier is defined by two complementary backflow-related architectures: the Sparse Convolutional Ansatz for Lattice Electrons (SCALE), which achieves high computational efficiency, and the Accurate Convolutional ansatz for lattice Electrons (ACE), which delivers high variational accuracy, based on benchmark results on the Hubbard model and the t – J model on large lattice sizes. SCALE utilizes a tailored convolutional design to enable an efficient local update based on the low-rank update of determinants. This structure reduces the computational scaling from O ( N 4 ) to O ( N 3 ) in backflow methods, resulting in a practical speedup exceeding 40 × in our test cases while retaining accurate variational accuracy. As an application, we study the 1 / 8 -doped pure Hubbard model up to size 32 × 32 and find no significant energy difference between the horizontal and vertical unit-filled stripe states, in contrast to the half-filled stripe state when next-nearest-neighbor hoppings are included. ACE, conversely, employs a deep convolutional stack to maximize expressive power, achieving high accuracy on large systems. We perform extensive benchmark calculations on the Hubbard and t – J models with the two architectures. On the challenging repulsive Hubbard model, SCALE achieves variational energies competitive with leading methods at a fraction of the computational cost. Meanwhile, ACE achieves lower variational energies than previously reported results while requiring only one-sixth of the run-time for system size 16 × 4 . These NQS approaches provide , , and tools for exploring the rich physics of strongly correlated fermionic systems.