# Investigating Topological Order using Recurrent Neural Networks

arXiv: https://arxiv.org/abs/2303.11207 · HTML: https://mlsanity.com/paper/2303.11207/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 18 of its 99 references that are in the map.
- First posted on arXiv 20 Mar 2023.

## Builds on (most cited first, 18 of 18)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Monte Carlo Gradient Estimation in Machine Learning](https://mlsanity.com/paper/1906.10652/) (2019)
- [A Critical Review of Recurrent Neural Networks for Sequence Learning](https://mlsanity.com/paper/1506.00019/) (2015)
- [Ab-Initio Solution of the Many-Electron Schr"odinger Equation with Deep Neural Networks](https://mlsanity.com/paper/1909.02487/) (2019)
- [Deep neural network solution of the electronic Schr"odinger equation](https://mlsanity.com/paper/1909.08423/) (2019)
- [Variational Neural Annealing](https://mlsanity.com/paper/2101.10154/) (2021)
- [Minimal Gated Unit for Recurrent Neural Networks](https://mlsanity.com/paper/1603.09420/) (2016)
- [Neural-Network Quantum States, String-Bond States, and Chiral Topological States](https://mlsanity.com/paper/1710.04045/) (2017)
- [Equivalence of restricted Boltzmann machines and tensor network states](https://mlsanity.com/paper/1701.04831/) (2017)
- [Locally Masked Convolution for Autoregressive Models](https://mlsanity.com/paper/2006.12486/) (2020)
- [Neural Error Mitigation of Near-Term Quantum Simulations](https://mlsanity.com/paper/2105.08086/) (2021)
- [Mutual Information Scaling and Expressive Power of Sequence Models](https://mlsanity.com/paper/1905.04271/) (2019)
- [From Tensor Network Quantum States to Tensorial Recurrent Neural Networks](https://mlsanity.com/paper/2206.12363/) (2022)
- [Machine learning discovery of new phases in programmable quantum simulator snapshots](https://mlsanity.com/paper/2112.10789/) (2021)
- [Dynamical large deviations of two-dimensional kinetically constrained models using a neural-network state ansatz](https://mlsanity.com/paper/2011.08657/) (2020)
- [Supplementing Recurrent Neural Networks with Annealing to Solve Combinatorial Optimization Problems](https://mlsanity.com/paper/2207.08189/) (2022)
- [Supplementing Recurrent Neural Network Wave Functions with Symmetry and Annealing to Improve Accuracy](https://mlsanity.com/paper/2207.14314/) (2022)
- [TensorNetwork on TensorFlow: A Spin Chain Application Using Tree Tensor Networks](https://mlsanity.com/paper/1905.01331/) (2019)

## Built on by (most cited first, 6 of 6)

- [Recurrent neural network wave functions for Rydberg atom arrays on kagome lattice](https://mlsanity.com/paper/2405.20384/) (2024)
- [When can classical neural networks represent quantum states?](https://mlsanity.com/paper/2410.23152/) (2024)
- [Tensorization of neural networks for improved privacy and interpretability](https://mlsanity.com/paper/2501.06300/) (2025)
- [Lattice Protein Folding with Variational Annealing](https://mlsanity.com/paper/2502.20632/) (2025)
- [Adaptive Neural Quantum States: A Recurrent Neural Network Perspective](https://mlsanity.com/paper/2507.18700/) (2025)
- [Large-scale portfolio optimization with variational neural annealing](https://mlsanity.com/paper/2507.07159/) (2025)

## Abstract (arXiv)

Recurrent neural networks (RNNs), originally developed for natural language processing, hold great promise for accurately describing strongly correlated quantum many-body systems. Here, we employ 2D RNNs to investigate two prototypical quantum many-body Hamiltonians exhibiting topological order. Specifically, we demonstrate that RNN wave functions can effectively capture the topological order of the toric code and a Bose-Hubbard spin liquid on the kagome lattice by estimating their topological entanglement entropies. We also find that RNNs favor coherent superpositions of minimally-entangled states over minimally-entangled states themselves. Overall, our findings demonstrate that RNN wave functions constitute a powerful tool to study phases of matter beyond Landau's symmetry-breaking paradigm.