# Equivalence of restricted Boltzmann machines and tensor network states

arXiv: https://arxiv.org/abs/1701.04831 · HTML: https://mlsanity.com/paper/1701.04831/

## Key facts

- Cited by 27 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 17 Jan 2017.

## Built on by (most cited first, 27 of 27)

- [Tensor Networks Meet Neural Networks: A Survey and Future Perspectives](https://mlsanity.com/paper/2302.09019/) (2023)
- [Grokking phase transitions in learning local rules with gradient descent](https://mlsanity.com/paper/2210.15435/) (2022)
- [Equilibrium and non-Equilibrium regimes in the learning of Restricted Boltzmann Machines](https://mlsanity.com/paper/2105.13889/) (2021)
- [Neural tensor contractions and the expressive power of deep neural quantum states](https://mlsanity.com/paper/2103.10293/) (2021)
- [Tensor networks for quantum machine learning](https://mlsanity.com/paper/2303.11735/) (2023)
- [Tensor networks for interpretable and efficient quantum-inspired machine learning](https://mlsanity.com/paper/2311.11258/) (2023)
- [Interpretable Machine Learning in Physics: A Review](https://mlsanity.com/paper/2503.23616/) (2025)
- [A Leap among Quantum Computing and Quantum Neural Networks: A Survey](https://mlsanity.com/paper/2107.03313/) (2021)
- [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)
- [Exponentially improved efficient machine learning for quantum many-body states with provable guarantees](https://mlsanity.com/paper/2304.04353/) (2023)
- [Investigating Topological Order using Recurrent Neural Networks](https://mlsanity.com/paper/2303.11207/) (2023)
- [Entangled q-Convolutional Neural Nets](https://mlsanity.com/paper/2103.11785/) (2021)
- [Estimating the Euclidean quantum propagator with deep generative modeling of Feynman paths](https://mlsanity.com/paper/2202.02750/) (2022)
- [Non-parametric Semi-Supervised Learning in Many-body Hilbert Space with Rescaled Logarithmic Fidelity](https://mlsanity.com/paper/2107.00195/) (2021)
- [Continuous-variable neural-network quantum states and the quantum rotor model](https://mlsanity.com/paper/2107.07105/) (2021)
- [Neural Tangent Kernel of Matrix Product States: Convergence and Applications](https://mlsanity.com/paper/2111.14046/) (2021)
- [Deep tensor networks with matrix product operators](https://mlsanity.com/paper/2209.09098/) (2022)
- [Representation Theorem for Matrix Product States](https://mlsanity.com/paper/2103.08277/) (2021)
- [Towards Neural Variational Monte Carlo That Scales Linearly with System Size](https://mlsanity.com/paper/2212.11296/) (2022)
- [Stack operation of tensor networks](https://mlsanity.com/paper/2203.16338/) (2022)
- [Neural network quantum state with proximal optimization: a ground-state searching scheme based on variational Monte Carlo](https://mlsanity.com/paper/2210.16493/) (2022)
- [Neural-network quantum state study of the long-range antiferromagnetic Ising chain](https://mlsanity.com/paper/2308.09709/) (2023)
- [Enhancing Deep Learning Models through Tensorization: A Comprehensive Survey and Framework](https://mlsanity.com/paper/2309.02428/) (2023)
- [Simulating Non-Markovian Open Quantum Dynamics with Neural Quantum States](https://mlsanity.com/paper/2404.11093/) (2024)
- [Seeding neural network quantum states with tensor network states](https://mlsanity.com/paper/2506.23550/) (2025)
- [Gaussian Universality and Its Breakdown in Tensor-Network Machine Learning](https://mlsanity.com/paper/2610.06080/) (2026)

## Abstract (arXiv)

The restricted Boltzmann machine (RBM) is one of the fundamental building blocks of deep learning. RBM finds wide applications in dimensional reduction, feature extraction, and recommender systems via modeling the probability distributions of a variety of input data including natural images, speech signals, and customer ratings, etc. We build a bridge between RBM and tensor network states (TNS) widely used in quantum many-body physics research. We devise efficient algorithms to translate an RBM into the commonly used TNS. Conversely, we give sufficient and necessary conditions to determine whether a TNS can be transformed into an RBM of given architectures. Revealing these general and constructive connections can cross-fertilize both deep learning and quantum many-body physics. Notably, by exploiting the entanglement entropy bound of TNS, we can rigorously quantify the expressive power of RBM on complex data sets. Insights into TNS and its entanglement capacity can guide the design of more powerful deep learning architectures. On the other hand, RBM can represent quantum many-body states with fewer parameters compared to TNS, which may allow more efficient classical simulations.