# Gauge equivariant neural networks for quantum lattice gauge theories

arXiv: https://arxiv.org/abs/2012.05232 · HTML: https://mlsanity.com/paper/2012.05232/

## Key facts

- Cited by 28 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 9 Dec 2020.

## Built on by (most cited first, 28 of 28)

- [Scalars are universal: Equivariant machine learning, structured like classical physics](https://mlsanity.com/paper/2106.06610/) (2021)
- [Geometric Deep Learning and Equivariant Neural Networks](https://mlsanity.com/paper/2105.13926/) (2021)
- [Introduction to Normalizing Flows for Lattice Field Theory](https://mlsanity.com/paper/2101.08176/) (2021)
- [QuanTA: Efficient High-Rank Fine-Tuning of LLMs with Quantum-Informed Tensor Adaptation](https://mlsanity.com/paper/2406.00132/) (2024)
- [Homogeneous vector bundles and G-equivariant convolutional neural networks](https://mlsanity.com/paper/2105.05400/) (2021)
- [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)
- [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)
- [ANTN: Bridging Autoregressive Neural Networks and Tensor Networks for Quantum Many-Body Simulation](https://mlsanity.com/paper/2304.01996/) (2023)
- [Algorithmic Differentiation for Automated Modeling of Machine Learned Force Fields](https://mlsanity.com/paper/2208.12104/) (2022)
- [Approximately-symmetric neural networks for quantum spin liquids](https://mlsanity.com/paper/2405.17541/) (2024)
- [Infinite Neural Network Quantum States: Entanglement and Training Dynamics](https://mlsanity.com/paper/2112.00723/) (2021)
- [Machine Learning in Nuclear Physics](https://mlsanity.com/paper/2112.02309/) (2021)
- [Why self-attention is Natural for Sequence-to-Sequence Problems? A Perspective from Symmetries](https://mlsanity.com/paper/2210.06741/) (2022)
- [Gauge-Equivariant Graph Neural Networks for Lattice Gauge Theories](https://mlsanity.com/paper/2604.20797/) (2026)
- [Continuous-variable neural-network quantum states and the quantum rotor model](https://mlsanity.com/paper/2107.07105/) (2021)
- [Densely Connected G-invariant Deep Neural Networks with Signed Permutation Representations](https://mlsanity.com/paper/2303.04614/) (2023)
- [Current Symmetry Group Equivariant Convolution Frameworks for Representation Learning](https://mlsanity.com/paper/2409.07327/) (2024)
- [Learning Chern Numbers of Topological Insulators with Gauge Equivariant Neural Networks](https://mlsanity.com/paper/2502.15376/) (2025)
- [SESaMo: Symmetry-Enforcing Stochastic Modulation for Normalizing Flows](https://mlsanity.com/paper/2505.19619/) (2025)
- [Graph Neural Networks in the Wilson Loop Representation of Abelian Lattice Gauge Theories](https://mlsanity.com/paper/2605.03901/) (2026)
- [Learning quantum symmetries with interactive quantum-classical variational algorithms](https://mlsanity.com/paper/2206.11970/) (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)
- [Holographic Quantum Transformer: A Generalist Neuro-Symbolic Architecture for Solving Frustrated Systems via Generative Attention](https://mlsanity.com/paper/2607.00398/) (2026)
- [Learning Discrete Riemannian Metrics for Physical Fields with Cochain-Frame Equivarianc](https://mlsanity.com/paper/2608.14556/) (2026)
- [Emergent Loop Geometry in Gauge-Equivariant Graph Neural Networks](https://mlsanity.com/paper/2610.18742/) (2026)

## Abstract (arXiv)

Gauge symmetries play a key role in physics appearing in areas such as quantum field theories of the fundamental particles and emergent degrees of freedom in quantum materials. Motivated by the desire to efficiently simulate many-body quantum systems with exact local gauge invariance, gauge equivariant neural-network quantum states are introduced, which exactly satisfy the local Hilbert space constraints necessary for the description of quantum lattice gauge theory with Zd gauge group on different geometries. Focusing on the special case of Z2 gauge group on a periodically identified square lattice, the equivariant architecture is analytically shown to contain the loop-gas solution as a special case. Gauge equivariant neural-network quantum states are used in combination with variational quantum Monte Carlo to obtain compact descriptions of the ground state wavefunction for the Z2 theory away from the exactly solvable limit, and to demonstrate the confining/deconfining phase transition of the Wilson loop order parameter.