Pareto Frontier of Neural Quantum States: Scalable, Affordable, and Accurate Convolutional Backflow for Strongly Correlated Lattice Fermions
Neural Quantum States (NQS) are now among the most accurate methods for studying strongly correlated many-fermion systems, outperforming existing many-body approaches for large systems. However, NQS calculations remain extremely resource-intensive. Here, we introduce a new Pareto frontier of efficiency and accuracy for NQS in simulating strongly correlated lattice fermions, defined by two complementary backflow-related architectures: the Sparse Convolutional Ansatz for Lattice Electrons (SCALE) (state-of-the-art efficiency) and the Accurate Convolutional ansatz for lattice Electrons (ACE) (state-of-the-art accuracy), benchmarked on the iconic Hubbard and $t-J$ models for large lattices. SCALE uses a tailored convolutional design enabling efficient local updates via low-rank determinant updates, reducing computational scaling from $O(N^4)$ to $O(N^3)$ in backflow methods and yielding a >40$\times$ practical speed-up in tests while maintaining high variational accuracy. As an application, we study the previously inaccessible 1/8-doped pure Hubbard model up to $32 \times 32$, finding no significant energy difference between horizontal and vertical filled stripe states - contrasting with half-filled stripe states when next-nearest-neighbor hoppings are included. ACE employs a deep convolutional stack to maximize expressive power, achieving unprecedented accuracy on large systems. Extensive benchmarks on Hubbard and $t-J$ models show SCALE delivers variational energies competitive with leading methods at a fraction of the cost, while ACE sets a new accuracy benchmark, surpassing recent results with only 1/6 the runtime for $16 \times 4$ systems. These new NQS approaches provide scalable, affordable, and accurate tools for exploring strongly correlated fermionic physics, such as the microscopic mechanism of unconventional superconductivity.