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arXiv:2506.08329 · Physical review. B./Physical review. B

Neuralized fermionic tensor networks for quantum many-body systems

Si-Jing Du, Ao Chen, Garnet Kin‐Lic Chan

Published Jan 22, 2026• 1 citation
arXivHTMLPDF

Abstract

We describe a class of neuralized fermionic tensor network states (NN-fTNS) that introduce non-linearity into fermionic tensor networks through configuration-dependent neural network transformations of the local tensors. The construction uses the fTNS algebra to implement a natural fermionic sign structure and is compatible with standard tensor network algorithms, but gains enhanced expressivity through the neural network parametrization. Using the 1D and 2D Fermi-Hubbard models as benchmarks, we demonstrate that NN-fTNS achieve order of magnitude improvements in the ground-state energy compared to pure fTNS with the same bond dimension, and can be systematically improved through both the tensor network bond dimension and the neural network parametrization. Compared to existing fermionic neural quantum states (NQS) based on Slater determinants and Pfaffians, NN-fTNS offer a physically motivated alternative fermionic structure. Furthermore, compared to such states, NN-fTNS naturally exhibit improved computational scaling and we demonstrate a construction that achieves linear scaling with the lattice size.

DOI
10.1103/x8vl-qf14

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•Aug 19, 2021•0 citations
We describe a class of neuralized fermionic tensor network states (NN-fTNS) that introduce non-linearity into fermionic tensor networks through configuration-dependent neural network transformations of the local tensors. The construction uses the fTNS algebra to implement a natural fermionic sign structure and is compatible with standard tensor network algorithms, but gains enhanced expressivity through the neural network parametrization. Using the 1D and 2D Fermi-Hubbard models as benchmarks, we demonstrate that NN-fTNS achieve order of magnitude improvements in the ground-state energy compared to pure fTNS with the same bond dimension, and can be systematically improved through both the tensor network bond dimension and the neural network parametrization. Compared to existing fermionic neural quantum states (NQS) based on Slater determinants and Pfaffians, NN-fTNS offer a physically motivated alternative fermionic structure. Furthermore, compared to such states, NN-fTNS naturally exhibit improved computational scaling and we demonstrate a construction that achieves linear scaling with the lattice size.
We introduce Neural Tensor Network States (νννTNS), a variational many-body wave-function ansatz that integrates deep neural networks with tensor-network architectures. In the νννTNS framework, a neural network serves as a disentangler of the wave-function, transforming the physical degrees of freedom into renormalized variables with much less entanglement. The renormalized state is then efficiently encoded by a back-flow tensor network. This construction yields a compact yet highly expressive representation of strongly correlated quantum states. Using convolutional neural networks combined with matrix product states as a concrete implementation, we obtain state-of-the-art variational energies for the spin-1/21/21/2 J1J_1J1​-J2J_2J2​ Heisenberg model on the square lattice at the highly frustrated point J2/J1=0.5J_2/J_1=0.5J2​/J1​=0.5, for systems up to 20×2020\times 2020×20 with periodic boundary conditions. Finite-size scaling of spin, dimer, and plaquette correlations exhibits power-law decay without magnetic or valence-bond long-range order, consistent with a gapless quantum spin-liquid ground state at that point.This νννTNS framework is flexible and naturally extensible to other neural and tensor-network structures, offering a general platform for investigating strongly correlated quantum many-body systems.
Abstract Numerically simulating large, spinful, fermionic systems is of great interest in condensed matter physics. However, the exponential growth of the Hilbert space dimension with system size renders exact quantum state parameterizations impractical. Owing to their representative power, neural networks often allow to overcome this exponential scaling. Here, we investigate the ability of neural quantum states (NQS) to represent the bosonic and fermionict − Jmodel – the high interaction limit of the Hubbard model – on various 1D and 2D lattices. Using autoregressive, tensorized recurrent neural networks (RNNs), we study ground state representations upon hole doping the half-filled system. Additionally, we propose a method to calculate quasiparticle dispersions, applicable to any network architecture or lattice geometry, and allowing to infer the low-energy physics from NQS. By analyzing the strengths and weaknesses of the RNN ansatz we shed light on the challenges and promises of NQS for simulating bosonic and fermionic systems.
The quantum many-body problem is an important topic in condensed matter physics. To efficiently solve the problem, several methods have been developed to improve the representation ability of wave functions. For the Fermi-Hubbard model under periodic boundary conditions, current state-of-the-art methods are neural network backflows and the hidden fermion Slater determinant. The backflow correction is an efficient way to improve the Slater determinant of free particles. In this work we propose a tensor representation of the backflow-corrected wave function; we show that for the spinless $t\text{\ensuremath{-}}V$ model, the energy precision is competitive or even lower than current state-of-the-art fermionic tensor network methods. For models with spin, we further improve the representation ability by considering backflows on fictitious particles with different spins, thus naturally introducing nonzero backflow corrections when the orbital and the particle have opposite spins. We benchmark our method on molecules in the STO-3G basis and the Fermi-Hubbard model with periodic and cylindrical boundary conditions. We show that the tensor representation of backflow corrections achieves competitive or even lower-energy results than current state-of-the-art neural network methods.
We establish a direct connection between general tensor networks and deep feed-forward artificial neural networks. The core of our results is the construction of neural-network layers that efficiently perform tensor contractions and that use commonly adopted nonlinear activation functions. The resulting deep networks feature a number of edges that closely match the contraction complexity of the tensor networks to be approximated. In the context of many-body quantum states, this result establishes that neural-network states have strictly the same or higher expressive power than practically usable variational tensor networks. As an example, we show that all matrix product states can be efficiently written as neural-network states with a number of edges polynomial in the bond dimension and depth that is logarithmic in the system size. The opposite instead does not hold true, and our results imply that there exist quantum states that are not efficiently expressible in terms of matrix product states or projected entangled pair states but that are instead efficiently expressible with neural network states.
We propose a general framework for finding the ground state of many-body fermionic systems by using feed-forward neural networks. The anticommutation relation for fermions is usually implemented to a variational wave function by the Slater determinant (or Pfaffian), which is a computational bottleneck because of the numerical cost of O(N3)O({N}^{3})O(N3) for NNN particles. We bypass this bottleneck by explicitly calculating the sign changes associated with particle exchanges in real space and using fully connected neural networks for optimizing the rest parts of the wave function. This reduces the computational cost from O(N3)O({N}^{3})O(N3) to O(N2)O({N}^{2})O(N2) or less, while the overall cost depends on the number of Monte Carlo samples and the number of parameters in neural networks. We show that the accuracy of the approximation can be improved by optimizing the ``variance'' of the energy simultaneously with the energy itself. We also find that a reweighting method in Monte Carlo sampling can stabilize the calculation. These improvements can be applied to other approaches based on variational Monte Carlo methods. Moreover, we show that the accuracy can be further improved by using the symmetry of the system, the representative states, data preprocessing, and an additional neural network implementing a generalized Gutzwiller-Jastrow factor. We demonstrate the efficiency of the method by applying it to a two-dimensional Hubbard model.
We propose a general framework for finding the ground state of many-body fermionic systems by using feed-forward neural networks. The anticommutation relation for fermions is usually implemented to a variational wave function by the Slater determinant (or Pfaffian), which is a computational bottleneck because of the numerical cost of O(N3)O(N^3)O(N3) for NNN particles. We bypass this bottleneck by explicitly calculating the sign changes associated with particle exchanges in real space and using fully connected neural networks for optimizing the rest parts of the wave function. This reduces the computational cost to O(N2)O(N^2)O(N2) or less. We show that the accuracy of the approximation can be improved by optimizing the variance of the energy simultaneously with the energy itself. We also find that a reweighting method in Monte Carlo sampling can stabilize the calculation. These improvements can be applied to other approaches based on variational Monte Carlo methods. Moreover, we show that the accuracy can be further improved by using the symmetry of the system, the representative states, and an additional neural network implementing a generalized Gutzwiller-Jastrow factor. We demonstrate the efficiency of the method by applying it to a two-dimensional Hubbard model.