# Neural-Network Quantum States, String-Bond States, and Chiral Topological States

arXiv: https://arxiv.org/abs/1710.04045 · HTML: https://mlsanity.com/paper/1710.04045/

## Key facts

- Cited by 31 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 11 Oct 2017.

## Built on by (most cited first, 30 of 31)

- [Provably efficient machine learning for quantum many-body problems](https://mlsanity.com/paper/2106.12627/) (2021)
- [NetKet 3: Machine Learning Toolbox for Many-Body Quantum Systems](https://mlsanity.com/paper/2112.10526/) (2021)
- [Improved machine learning algorithm for predicting ground state properties](https://mlsanity.com/paper/2301.13169/) (2023)
- [Neural tensor contractions and the expressive power of deep neural quantum states](https://mlsanity.com/paper/2103.10293/) (2021)
- [Tensor networks for unsupervised machine learning](https://mlsanity.com/paper/2106.12974/) (2021)
- [Gauge Invariant and Anyonic Symmetric Transformer and RNN Quantum States for Quantum Lattice Models](https://mlsanity.com/paper/2101.07243/) (2021)
- [Tensor networks for interpretable and efficient quantum-inspired machine learning](https://mlsanity.com/paper/2311.11258/) (2023)
- [Gauge Equivariant Neural Networks for 2+1D U(1) Gauge Theory Simulations in Hamiltonian Formulation](https://mlsanity.com/paper/2211.03198/) (2022)
- [Simulating 2+1D Lattice Quantum Electrodynamics at Finite Density with Neural Flow Wavefunctions](https://mlsanity.com/paper/2212.06835/) (2022)
- [Pairing-based graph neural network for simulating quantum materials](https://mlsanity.com/paper/2311.02143/) (2023)
- [Can Theoretical Physics Research Benefit from Language Agents?](https://mlsanity.com/paper/2506.06214/) (2025)
- [NNQS-Transformer: an Efficient and Scalable Neural Network Quantum States Approach for Ab initio Quantum Chemistry](https://mlsanity.com/paper/2306.16705/) (2023)
- [Boltzmann machines as two-dimensional tensor networks](https://mlsanity.com/paper/2105.04130/) (2021)
- [Lattice Convolutional Networks for Learning Ground States of Quantum Many-Body Systems](https://mlsanity.com/paper/2206.07370/) (2022)
- [Investigating Topological Order using Recurrent Neural Networks](https://mlsanity.com/paper/2303.11207/) (2023)
- [Are queries and keys always relevant? A case study on Transformer wave functions](https://mlsanity.com/paper/2405.18874/) (2024)
- [Approximately-symmetric neural networks for quantum spin liquids](https://mlsanity.com/paper/2405.17541/) (2024)
- [Entangled q-Convolutional Neural Nets](https://mlsanity.com/paper/2103.11785/) (2021)
- [Scaling of neural-network quantum states for time evolution](https://mlsanity.com/paper/2104.10696/) (2021)
- [A simple framework for contrastive learning phases of matter](https://mlsanity.com/paper/2205.05607/) (2022)
- [Convergence Analysis of Stochastic Gradient Descent with MCMC Estimators](https://mlsanity.com/paper/2303.10599/) (2023)
- [Optical Quantum Mixed-State Reconstruction With Multiple Deep Learning Approaches](https://mlsanity.com/paper/2407.01734/) (2024)
- [Learning entanglement breakdown as a phase transition by confusion](https://mlsanity.com/paper/2202.00348/) (2022)
- [Three Learning Stages and Accuracy-Efficiency Tradeoff of Restricted Boltzmann Machines](https://mlsanity.com/paper/2209.00873/) (2022)
- [Predicting Quantum Potentials by Deep Neural Network and Metropolis Sampling](https://mlsanity.com/paper/2106.03126/) (2021)
- [Towards Neural Variational Monte Carlo That Scales Linearly with System Size](https://mlsanity.com/paper/2212.11296/) (2022)
- [Artificial intelligence for artificial materials: moir'e atom](https://mlsanity.com/paper/2303.08162/) (2023)
- [Mechanistic Interpretability and Causal Feature Steering of Neural Quantum States via Sparse Autoencoders](https://mlsanity.com/paper/2607.01336/) (2026)
- [Machine Learning Regression for Operator Dynamics](https://mlsanity.com/paper/2102.11868/) (2021)
- [Neural-network quantum state study of the long-range antiferromagnetic Ising chain](https://mlsanity.com/paper/2308.09709/) (2023)

## Abstract (arXiv)

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.