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 $4\times 16$ and $4 \times 8$ 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.
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 4×16 and 4×8 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.
The ground state of second-quantized quantum chemistry Hamiltonians provides access to an important set of chemical properties. Wave functions based on machine-learning architectures have shown promise in approximating these ground states in a variety of physical systems. In this paper, we show how to achieve state-of-the-art energies for molecular Hamiltonians using the the neural network backflow (NNBF) wave function. To accomplish this, we optimize this ansatz with a variant of the deterministic optimization scheme based on selected configuration interaction introduced by Li et al., [J. Chem. Theory Comput. 19, 8156 (2023)], which we find works better than standard Markov chain Monte Carlo sampling. For the molecules we studied, NNBF gives lower energy states than both Coupled Cluster with Single and Double excitations and other neural network quantum states. We systematically explore the role of network size as well as optimization parameters in improving the energy. We find that, while the number of hidden layers and determinants play a minor role in improving the energy, there are significant improvements in the energy from increasing the number of hidden units as well as the batch size used in optimization, with the batch size playing a more important role.
Obtaining an accurate ground state wave function is one of the great challenges in the quantum many-body problem. In this Letter, we propose a new class of wave functions, neural network backflow (NNB). The backflow approach, pioneered originally by Feynman and Cohen [Phys. Rev. 102, 1189 (1956)10.1103/PhysRev.102.1189], adds correlation to a mean-field ground state by transforming the single-particle orbitals in a configuration-dependent way. NNB uses a feed-forward neural network to learn the optimal transformation via variational Monte Carlo calculations. NNB directly dresses a mean-field state, can be systematically improved, and directly alters the sign structure of the wave function. It generalizes the standard backflow [L. F. Tocchio et al., Phys. Rev. B 78, 041101(R) (2008)10.1103/PhysRevB.78.041101], which we show how to explicitly represent as a NNB. We benchmark the NNB on Hubbard models at intermediate doping, finding that it significantly decreases the relative error, restores the symmetry of both observables and single-particle orbitals, and decreases the double-occupancy density. Finally, we illustrate interesting patterns in the weights and bias of the optimized neural network.
Among the variational wave functions for fermionic Hamiltonians, neural network backflow (NNBF) and hidden fermion determinant states (HFDS) are two prominent classes that provide accurate approximations to the ground state. Here we develop a unifying view of fermionic neural quantum states casting them all in the framework of NNBF. NNBF wave functions have configuration-dependent single-particle orbitals (SPO) which are parameterized by a neural network. We show that HFDS can be written as a NNBF wave function with a restricted low-rank r additive correction to the SPO times a neural-network generated determinant of an $r\ifmmode\times\else\texttimes\fi{}r$ matrix. Furthermore, we show that in NNBF wave functions, this $r\ifmmode\times\else\texttimes\fi{}r$ determinant can generically be removed when r is less than or equal to the number of fermions, at the cost of further complicating the additive SPO correction increasing its rank by r. We numerically and analytically compare additive SPO corrections generated by the product of two matrices with inner dimension r. We find that larger r wave functions span a larger space and give evidence that simpler and more direct updates to the SPO's tend to be more expressive and better energetically. These suggest the standard NNBF approach is preferred amongst other related choices. Finally, we uncover that the row selection used to select single-particle orbitals allows significant sign and amplitude modulation between nearby configurations and is partially responsible for the quality of NNBF and HFDS wave functions.
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.
Strongly correlated molecules often contain dense manifolds of low-lying spin states, making total-spin symmetry essential for predictive electronic-structure theory. Neural-network quantum states provide flexible variational wavefunctions, but commonly used fermionic architectures do not enforce this symmetry and can therefore converge to spin-contaminated states with misleading energies and properties. Here we introduce a spin-adapted neural-network backflow (SA-NNBF) ansatz in second quantization, which combines configuration-dependent spatial orbitals with a compressed spin eigenfunction. A projected tensor compression scheme for spin eigenfunctions and a particle-hole representation make variational Monte Carlo calculations with SA-NNBF practical for active spaces containing more than one hundred electrons. Across hydrogen chains and iron-sulfur clusters, SA-NNBF eliminates spin contamination and consistently achieves lower variational energies than standard NNBF with a comparable number of parameters. For the CAS(113e,76o) active-space model of FeMoco, SA-NNBF yields a highly compact spin-adapted variational state, achieving an energy competitive with recent spin-adapted DMRG calculations at bond dimension D=10000 while using orders of magnitude fewer parameters. Our work establishes a general framework for developing spin-symmetry-preserving neural-network quantum states for chemically realistic strongly correlated electrons.
Accurately simulating extended periodic systems is a central challenge in condensed matter physics. Neural quantum states (NQS) offer expressive wavefunctions for this task but face issues with scalability. In this work, we successfully extend the neural network backflow (NNBF) approach to ab-initio solid-state materials. Building on our scalable optimization framework for molecules [Liu et al., PRB 112, 155162 (2025)], we introduce a two-stage pruning strategy to manage the massive configuration space expansions: by utilizing a computationally cheap, physics-informed importance proxy, we devote exact NNBF amplitude evaluations solely to the most relevant determinants, significantly improving optimization efficiency, energy estimation, and convergence. Our framework achieves state-of-the-art accuracy across diverse solid-state benchmarks. For 1D hydrogen chains, NNBF matches or surpasses DMRG and AFQMC, remains robust in strongly correlated bond-breaking regimes where coupled-cluster methods fail, and smoothly extrapolates to the thermodynamic limit. We further demonstrate its scalability by computing ground-state potential energy curves for 2D graphene and 3D silicon. Finally, ablation studies validate the computational savings of our pruning strategy and highlight the dependence of the NNBF energies on basis sets.