Neuralized Fermionic Tensor Networks for Quantum Many-Body Systems

TL;DR

This work introduces NN-fTNS, a class of neuralized fermionic tensor network states that inject nonlinearity into local tensors while preserving the fermionic sign structure via a $\mathbb{Z}_2$-graded tensor product. By coupling a self-attention–plus–MLP neural block to each on-site tensor, NN-fTNS creates configuration-dependent amplitudes $\langle n|\Psi(n)\rangle$ that can be contracted with existing fTNS algorithms. Across 1D and 2D Fermi-Hubbard benchmarks, NN-fTNS consistently achieve substantially lower ground-state energies than pure fTNS and Slater-determinant–based NQS, including cases where an $D=8$ NN-fTNS beats a pure $D=16$ fPEPS, demonstrating improved expressivity and favorable scaling. The work also shows that restricting neural dependence to a finite range $R$ enables partial contraction reuse, offering a path toward near-linear scaling with system size while maintaining high accuracy.

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.

Figures (10)

• Figure 1: (a) A fermionic PEPS, where the arrows indicate the fermionic nature of the tensor legs Mortier2025gao2024. (b) Illustration of the boundary MPS method used to define the contraction of an amplitude fPEPS. (c) Sketch of the computation of amplitudes $\langle n|\Psi\rangle$ using the NN-fTNS Ansatz.
• Figure 2: Relative energy error versus the number of variational parameters for various relevant Ansätze on the $4\times 2$ FH model at hole doping $n_h=1/4$. Errors are calculated relative to the exact ground state energy. NN: pure neural network parametrization; NN-SD: optimized Slater determinant with NN backflow (specific NN structure indicated in parentheses); fPEPS: pure fPEPS state; NN-fPEPS: neuralized fPEPS; fPEPS$\times$NN-Jastrow: Multiplicative NN-Jastrow factor on fPEPS.
• Figure 3: Relative energy error w.r.t. DMRG energy of the FH model for different fTNS bond dimension $D$ with various lattice geometries and hole dopings. (a) $24$-site chain at hole doping $n_h=1/6$. (b) $4\times 4$ square lattice at half filling. (c) $4\times 4$ square lattice at $n_h=1/8$ hole doping. (d) $6\times 6$ square lattice at $n_h=1/9$ hole doping. DMRG energy for the one-dimensional FH model is obtained with a fMPS of bond dimension $m=1000$. DMRG and fPEPS energies for the two-dimensional FHM are taken from Ref. Liu2025. In (b) and (c), we provide the baseline energies of NN-SD calculations with sophisticated model architectures for comparison. The NN-SD baseline models each have about 100,000 parameters, a size comparable to that of the NN-fPEPS model with $D=6$.
• Figure 4: Impact of the neural network layer width $W$ on the energy accuracy of NN-fPEPS for different fPEPS bond dimension $D$. Hamiltonian: FH model at $n_h=1/8$ hole doping on a $4\times 4$ square lattice. Errors are calculated relative to DMRG energy in Liu2025.
• Figure 5: (a) Time cost comparison for a single VMC step ($20$ samples per step) across different fPEPS bond dimensions $D$ on a $6 \times 6$ FH model. Each timing is averaged over 5 independent trials on a single CPU core. The boundary MPS contraction bond dimension is chosen as $\chi = 4D$. (b) Scalings of the time cost per VMC step ($20$ samples per step) with system size $N$, using NN-fMPS ($D=6$) models with different neural network range $R$. Each timing is averaged over 5 independent trials on a single CPU core. (c)-(d): Relative ground-state energy errors of NN-fTNS with varying neural network ranges $R$ for the FH model on a (c) $24$-site chain and (d) $6\times 6$ square lattice. In (d), the fPEPS benchmark energy is taken from Ref. Liu2025.