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arXiv:2603.14425

Disentangling Tensor Network States with Deep Neural Network

Chaohui Fan, Bo Zhan, Yuntian Gu, Tong Liu, Yantao Wu, Mingpu Qin, Dingshun Lv, Tao Xiang

cond-mat.str-elquant-phSubmitted Mar 15, 2026• Announced Mar 17, 2026• 0 citations
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Abstract

We introduce Neural Tensor Network States ($ν$TNS), a variational many-body wave-function ansatz that integrates deep neural networks with tensor-network architectures. In the $ν$TNS 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-$1/2$ $J_1$-$J_2$ Heisenberg model on the square lattice at the highly frustrated point $J_2/J_1=0.5$, for systems up to $20\times 20$ 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 $ν$TNS 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.

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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.
We show that neural quantum states based on very deep (4--16-layered) neural networks can outperform state-of-the-art variational approaches on highly frustrated quantum magnets, including quantum-spin-liquid candidates. We focus on group convolutional neural networks (GCNNs) that allow us to impose space-group symmetries on our ansätze. We achieve state-of-the-art ground-state energies for the J1−J2J_1-J_2J1​−J2​ Heisenberg models on the square and triangular lattices, in both ordered and spin-liquid phases, and discuss ways to access low-lying excited states in nontrivial symmetry sectors. We also compute spin and dimer correlation functions for the quantum paramagnetic phase on the triangular lattice, which do not indicate either conventional or valence-bond ordering.
In this chapter, we describe a tensor network (TN) based common language established between machine learning and many-body physics, which allows for bidirectional contributions. By showing that many-body wave functions are structurally equivalent to mappings of convolutional and recurrent networks, we bring forth quantum entanglement measures as natural quantifiers of dependencies modeled by such networks. Accordingly, we propose a novel entanglement-based deep learning design scheme that sheds light on the success of popular architectural choices made by deep learning practitioners and suggests new practical prescriptions. In the other direction, we construct TNs corresponding to deep recurrent and convolutional networks. This allows us to theoretically demonstrate that these architectures are powerful enough to represent highly entangled quantum systems polynomially more efficiently than previously employed architectures. We thus provide theoretical motivation to shift neural-network-based wave function representations closer to state-of-the-art deep learning architectures.
By taking inspiration from the backflow transformation for correlated systems, we introduce a tensor network Ansatz which extends the well-established matrix product state representation of a quantum many-body wave function. This structure provides enough resources to ensure that states in dimensions larger than or equal to one obey an area law for entanglement. It can be efficiently manipulated to address the ground-state search problem by means of an optimization scheme which mixes tensor-network and variational Monte Carlo algorithms. We benchmark the Ansatz against spin models both in one and two dimensions, demonstrating high accuracy and precision. We finally employ our approach to study the challenging S=1/2S=1/2S=1/2 two-dimensional (2D) ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ model, demonstrating that it is competitive with the state-of-the-art methods in 2D.
The use of artificial neural networks to represent quantum wave functions has recently attracted interest as a way to solve complex many-body problems. The potential of these variational parametrizations has been supported by analytical and numerical evidence in controlled benchmarks. While approaching the end of the early research phase in this field, it becomes increasingly important to show how neural-network states perform for models and physical problems that constitute a clear open challenge for other many-body computational methods. In this paper, we start addressing this aspect, concentrating on a presently unsolved model describing two-dimensional frustrated magnets. Using a fully convolutional neural network model as a variational ans\"atz, we study the frustrated spin-1/$2{J}_{1}\text{\ensuremath{-}}{J}_{2}$ Heisenberg model on the square lattice. We demonstrate that the resulting predictions for both ground-state energies and properties are competitive with, and often improve upon, existing state-of-the-art methods. In a relatively small region in the parameter space, corresponding to the maximally frustrated regime, our ans\"atz exhibits comparatively good but not the best performance. The gap between the complexity of the models adopted here and those routinely adopted in deep-learning applications is, however, still substantial, such that further improvements in future generations of neural-network quantum states are likely to be expected.
Understanding strongly correlated quantum many-body systems requires dealing with a largeconfiguration space comprised of a set of all possible measurable configurations. The total number of configurations grows exponentially with the size of the system, which makes exact numerical calculations impossible for even moderately sized systems. Since the advent of high-temperature superconductivity and the subsequent interest in strongly correlated phases of matter, progress in the field has been marked by ingenuity of the field to overcome these computational limitations. The first techniques to emerge were inspired by quantum chemistry and based on exact diagonalization[1], which is limited to small clusters, and different variants of quantum Monte Carlo[2], that suffers from the sign problem in fermionic and frustrated systems. Circa 1992, the density matrix renormalization group (DMRG) [3, 4, 5, 6, 7] became one of the major breakthroughs of the past few decades. Despite the success of DMRG for one-dimensional(1D) and quasi-one-dimensional geometries, extensions to actual two-dimensional systems remain challenging and applications are constrained to long cylinders and strips. One promising alternative is to combine one the oldest methods, variational wave functions (VWFs), with neural networks, which have been used to recently with great success in machine learning. The focus of this dissertation is on developing novel variational Monte Carlo algorithms to make neural network variational wave functions applicable to many problems in quantum many-body physics. These algorithms draw from existing variational wave-function, tensor network, and machine learning methods. We calculate dynamical spectral functions using the vector correction and Chebyshev expansion method. We do finite temperature calculations with minimally entangled thermal states. Finally, we do ground state calculations for the frustrated J1−J2 Heisenberg model. We benchmark these results using restricted Boltzmann machines as the neural network for the the variational wave-functions.--Author's abstract
Neural-Network Quantum States have been recently introduced as an Ansatz for describing the wave function of quantum many-body systems. We show that there are strong connections between Neural-Network Quantum States in the form of Restricted Boltzmann Machines and some classes of Tensor-Network states in arbitrary dimensions. In particular we demonstrate that short-range Restricted Boltzmann Machines are Entangled Plaquette States, while fully connected Restricted Boltzmann Machines are String-Bond States with a nonlocal geometry and low bond dimension. These results shed light on the underlying architecture of Restricted Boltzmann Machines and their efficiency at representing many-body quantum states. String-Bond States also provide a generic way of enhancing the power of Neural-Network Quantum States and a natural generalization to systems with larger local Hilbert space. We compare the advantages and drawbacks of these different classes of states and present a method to combine them together. This allows us to benefit from both the entanglement structure of Tensor Networks and the efficiency of Neural-Network Quantum States into a single Ansatz capable of targeting the wave function of strongly correlated systems. While it remains a challenge to describe states with chiral topological order using traditional Tensor Networks, we show that Neural-Network Quantum States and their String-Bond States extension can describe a lattice Fractional Quantum Hall state exactly. In addition, we provide numerical evidence that Neural-Network Quantum States can approximate a chiral spin liquid with better accuracy than Entangled Plaquette States and local String-Bond States. Our results demonstrate the efficiency of neural networks to describe complex quantum wave functions and pave the way towards the use of String-Bond States as a tool in more traditional machine-learning applications.