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arXiv:2604.25775 · PRX Intelligence

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

Yuntian Gu, Zeyao Han, Wenrui Li, Zhiyu Xiao, Tao Xiang, Mingpu Qin, Liwei Wang, Dingshun Lv

Published Oct 6, 2026• 0 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/2kq2-5g5q

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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.
Abstract 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 cutting-edge transformer-based architectures and developing highly 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 and entanglements in strongly correlated systems. With these advances, we establish 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.
Accurate simulations of the Hubbard model are crucial to understanding strongly correlated phenomena, where 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 accuracy is similar. We also test different symmetrization strategies, finding that output averaging yields the lowest energies, though 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.
Neural quantum states provide a flexible and scalable framework for strongly correlated systems. This thesis develops and benchmarks variational neural ansÀtze for fermionic lattice models, targeting ground states and low-lying excited states, with a particular focus on architectural performance. The constructions embed known structure such as fermionic antisymmetry, lattice translations, and total momentum while learning the remaining correlations efficiently in log amplitude. In the first part, we introduce a single dimensionless, intensive accuracy indicator, the V-score. Defined from the mean energy and energy variance of a state, it enables objective, method-agnostic comparisons across system sizes, interaction strengths, and even distinct momentum sectors, including regimes without exact references. We use it as a unifying measure of variational quality throughout the thesis. In the second part, we introduce a symmetry-aware Slaterâ Backflowâ Jastrow ansatz for spinless fermions in the t-V model. A convolutional backflow is built to be translation equivariant and consistent with particle exchange, allowing quantum-number projection to a fixed momentum without additional forward passes. This state is used to target ground states and momentum-resolved excitations, and to probe ordering tendencies via structure factors. In the third part, we develop a momentum-projected vision-transformer ansatz (DefeViT) for the one-hole t-J family. The state is factorized in a fixed-hole frame with a neural amplitude for the mobile defect, then restored to a chosen total momentum by summing lattice translations with the appropriate fermionic phase. On square lattices and their diagonal extensions (t'-J'), the approach captures ground-state properties and single-hole dispersions and tracks how spin correlators and hole-centered bond textures evolve under frustration. Within this setting, physics-informed neural states provide compact and accurate descriptions of ground states and low-lying excited states.
Simulating strongly correlated fermions remains a long-standing challenge due to the exponential complexity of the Hilbert space and the intricate sign structure of many-body wavefunctions. We introduce a variational framework centered on a neural network backflow transformation that combines deep learning with variational Monte Carlo. The proposed ansatz employs a multi-determinant expansion with low-rank shifts to capture non-local correlations and complex sign structures. Applied to the two-dimensional Hubbard model at both half-filling and 1/81/81/8 doping, the method achieves energies within 0.45%0.45\%0.45% of auxiliary-field quantum Monte Carlo at half-filling and captures intertwined charge- and spin-density stripe patterns at 1/81/81/8 doping. These results demonstrate the potential of this framework as a scalable and interpretable approach to variational simulations of strongly correlated fermionic systems.
The ground state of second-quantized quantum chemistry Hamiltonians is key to determining molecular properties. Neural quantum states (NQS) offer flexible and expressive wavefunction ansatze for this task but face two main challenges: highly peaked ground-state wavefunctions hinder efficient sampling, and local energy evaluations scale quartically with system size, incurring significant computational costs. In this work, we overcome these challenges by introducing a suite of algorithmic enhancements, which includes efficient periodic compact subspace construction, truncated local energy evaluations, improved stochastic sampling, and physics-informed modifications. Applying these techniques to the neural network backflow (NNBF) ansatz, we demonstrate significant gains in both accuracy and scalability. Our enhanced method surpasses traditional quantum chemistry methods like CCSD and CCSD(T), outperforms other NQS approaches, and achieves competitive energies with state-of-the-art ab initio techniques such as HCI, ASCI, FCIQMC, and DMRG. A series of ablation and comparative studies quantifies the contribution of each enhancement to the observed improvements in accuracy and efficiency. Furthermore, we investigate the representational capacity of the ansatz, finding that its performance correlates with the inverse participation ratio (IPR), with more delocalized states being more challenging to approximate.
Accurately describing the ground state of strongly correlated systems is essential for understanding their emergent properties. Neural Network Backflow (NNBF) is a powerful variational ansatz that enhances mean-field wave functions by introducing configuration-dependent modifications to single-particle orbitals. Although NNBF is theoretically universal in the limit of large networks, we find that practical gains saturate with increasing network size. Instead, significant improvements can be achieved by using a multi-determinant ansatz. We explore efficient ways to generate these multi-determinant expansions without increasing the number of variational parameters. In particular, we study single-step Lanczos and symmetry projection techniques, benchmarking their performance against diffusion Monte Carlo and NNBF applied to alternative mean fields. Benchmarking on a doped periodic square Hubbard model near optimal doping, we find that a Lanczos step, diffusion Monte Carlo, and projection onto a symmetry sector all give similar improvements achieving state-of-the-art energies at minimal cost. By further optimizing the projected symmetrized states directly, we gain significantly in energy. Using this technique we report the lowest variational energies for this Hamiltonian on 4×164\times 164×16 and 4×84 \times 84×8 lattices as well as accurate variance extrapolated energies. We also show the evolution of spin, charge, and pair correlation functions as the quality of the variational ansatz improves.