# Approximately-symmetric neural networks for quantum spin liquids

arXiv: https://arxiv.org/abs/2405.17541 · HTML: https://mlsanity.com/paper/2405.17541/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 27 of its 113 references that are in the map.
- First posted on arXiv 27 May 2024.

## Builds on (most cited first, 27 of 27)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Deep Sets](https://mlsanity.com/paper/1703.06114/) (2017)
- [Group Equivariant Convolutional Networks](https://mlsanity.com/paper/1602.07576/) (2016)
- [Fast and Accurate Deep Network Learning by Exponential Linear Units (ELUs)](https://mlsanity.com/paper/1511.07289/) (2015)
- [General E(2)-Equivariant Steerable CNNs](https://mlsanity.com/paper/1911.08251/) (2019)
- [3D Steerable CNNs: Learning Rotationally Equivariant Features in Volumetric Data](https://mlsanity.com/paper/1807.02547/) (2018)
- [Steerable CNNs](https://mlsanity.com/paper/1612.08498/) (2016)
- [A Practical Method for Constructing Equivariant Multilayer Perceptrons for Arbitrary Matrix Groups](https://mlsanity.com/paper/2104.09459/) (2021)
- [Clebsch-Gordan Nets: a Fully Fourier Space Spherical Convolutional Neural Network](https://mlsanity.com/paper/1806.09231/) (2018)
- [Equivariance with Learned Canonicalization Functions](https://mlsanity.com/paper/2211.06489/) (2022)
- [Quantum Natural Gradient](https://mlsanity.com/paper/1909.02108/) (2019)
- [Approximately Equivariant Networks for Imperfectly Symmetric Dynamics](https://mlsanity.com/paper/2201.11969/) (2022)
- [Rank-normalization, folding, and localization: An improved widehatR for assessing convergence of MCMC](https://mlsanity.com/paper/1903.08008/) (2019)
- [Residual Pathway Priors for Soft Equivariance Constraints](https://mlsanity.com/paper/2112.01388/) (2021)
- [Tunable Efficient Unitary Neural Networks (EUNN) and their application to RNNs](https://mlsanity.com/paper/1612.05231/) (2016)
- [Deep autoregressive models for the efficient variational simulation of many-body quantum systems](https://mlsanity.com/paper/1902.04057/) (2019)
- [Lattice gauge equivariant convolutional neural networks](https://mlsanity.com/paper/2012.12901/) (2020)
- [Neural-Network Quantum States, String-Bond States, and Chiral Topological States](https://mlsanity.com/paper/1710.04045/) (2017)
- [Gauge equivariant neural networks for quantum lattice gauge theories](https://mlsanity.com/paper/2012.05232/) (2020)
- [NetKet 3: Machine Learning Toolbox for Many-Body Quantum Systems](https://mlsanity.com/paper/2112.10526/) (2021)
- [Symmetry Breaking and Equivariant Neural Networks](https://mlsanity.com/paper/2312.09016/) (2023)
- [Neural tensor contractions and the expressive power of deep neural quantum states](https://mlsanity.com/paper/2103.10293/) (2021)
- [Neural network wave functions and the sign problem](https://mlsanity.com/paper/2002.04613/) (2020)
- [Gauge Invariant and Anyonic Symmetric Transformer and RNN Quantum States for Quantum Lattice Models](https://mlsanity.com/paper/2101.07243/) (2021)
- [Gauge Equivariant Neural Networks for 2+1D U(1) Gauge Theory Simulations in Hamiltonian Formulation](https://mlsanity.com/paper/2211.03198/) (2022)
- [Learning to Convolve: A Generalized Weight-Tying Approach](https://mlsanity.com/paper/1905.04663/) (2019)
- [ANTN: Bridging Autoregressive Neural Networks and Tensor Networks for Quantum Many-Body Simulation](https://mlsanity.com/paper/2304.01996/) (2023)

## Built on by (most cited first, 5 of 5)

- [Interpretable Machine Learning in Physics: A Review](https://mlsanity.com/paper/2503.23616/) (2025)
- [Relaxed Equivariance via Multitask Learning](https://mlsanity.com/paper/2410.17878/) (2024)
- [Achieving Approximate Symmetry Is Exponentially Easier than Exact Symmetry](https://mlsanity.com/paper/2512.11855/) (2025)
- [Learning topological states from randomized measurements using variational tensor network tomography](https://mlsanity.com/paper/2406.00193/) (2024)
- [Mechanistic Interpretability and Causal Feature Steering of Neural Quantum States via Sparse Autoencoders](https://mlsanity.com/paper/2607.01336/) (2026)

## Abstract (arXiv)

We propose and analyze a family of approximately-symmetric neural networks for quantum spin liquid problems. These tailored architectures are parameter-efficient, scalable, and significantly outperform existing symmetry-unaware neural network architectures. Utilizing the mixed-field toric code and PXP Rydberg Hamiltonian models, we demonstrate that our approach is competitive with the state-of-the-art tensor network and quantum Monte Carlo methods. Moreover, at the largest system sizes (N = 480 for toric code, N=1584 for Rydberg PXP), our method allows us to explore Hamiltonians with sign problems beyond the reach of both quantum Monte Carlo and finite-size matrix-product states. The network comprises an exactly symmetric block following a non-symmetric block, which we argue learns a transformation of the ground state analogous to quasiadiabatic continuation. Our work paves the way toward investigating quantum spin liquid problems within interpretable neural network architectures.