# Equivariant quantum circuits for learning on weighted graphs

arXiv: https://arxiv.org/abs/2205.06109 · HTML: https://mlsanity.com/paper/2205.06109/

## Key facts

- Cited by 41 papers in the map (data through 14 Oct 2026).
- Builds on 28 of its 69 references that are in the map.
- First posted on arXiv 12 May 2022.

## Builds on (most cited first, 28 of 28)

- [Zero-Shot Text-to-Image Generation](https://mlsanity.com/paper/2102.12092/) (2021)
- [A Comprehensive Survey on Graph Neural Networks](https://mlsanity.com/paper/1901.00596/) (2019)
- [Graph Neural Networks: A Review of Methods and Applications](https://mlsanity.com/paper/1812.08434/) (2018)
- [Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges](https://mlsanity.com/paper/2104.13478/) (2021)
- [Pointer Networks](https://mlsanity.com/paper/1506.03134/) (2015)
- [Graph Neural Networks for Social Recommendation](https://mlsanity.com/paper/1902.07243/) (2019)
- [Variational Quantum Algorithms](https://mlsanity.com/paper/2012.09265/) (2020)
- [Barren plateaus in quantum neural network training landscapes](https://mlsanity.com/paper/1803.11173/) (2018)
- [Machine Learning for Combinatorial Optimization: a Methodological Tour d'Horizon](https://mlsanity.com/paper/1811.06128/) (2018)
- [Learning Combinatorial Optimization Algorithms over Graphs](https://mlsanity.com/paper/1704.01665/) (2017)
- [Reinforcement Learning for Solving the Vehicle Routing Problem](https://mlsanity.com/paper/1802.04240/) (2018)
- [Parameterized quantum circuits as machine learning models](https://mlsanity.com/paper/1906.07682/) (2019)
- [Combinatorial optimization and reasoning with graph neural networks](https://mlsanity.com/paper/2102.09544/) (2021)
- [Generalization in quantum machine learning from few training data](https://mlsanity.com/paper/2111.05292/) (2021)
- [Ising formulations of many NP problems](https://mlsanity.com/paper/1302.5843/) (2013)
- [Noise-Induced Barren Plateaus in Variational Quantum Algorithms](https://mlsanity.com/paper/2007.14384/) (2020)
- [Variational Quantum Circuits for Deep Reinforcement Learning](https://mlsanity.com/paper/1907.00397/) (2019)
- [Entanglement Induced Barren Plateaus](https://mlsanity.com/paper/2010.15968/) (2020)
- [Molecular geometry prediction using a deep generative graph neural network](https://mlsanity.com/paper/1904.00314/) (2019)
- [Learning with invariances in random features and kernel models](https://mlsanity.com/paper/2102.13219/) (2021)
- [Chip Placement with Deep Reinforcement Learning](https://mlsanity.com/paper/2004.10746/) (2020)
- [Effect of barren plateaus on gradient-free optimization](https://mlsanity.com/paper/2011.12245/) (2020)
- [Reinforcement Learning with Quantum Variational Circuits](https://mlsanity.com/paper/2008.07524/) (2020)
- [Recurrent Neural Networks (RNNs): A gentle Introduction and Overview](https://mlsanity.com/paper/1912.05911/) (2019)
- [Quantum Graph Neural Networks](https://mlsanity.com/paper/1909.12264/) (2019)
- [Group-Invariant Quantum Machine Learning](https://mlsanity.com/paper/2205.02261/) (2022)
- [Quantum Graph Convolutional Neural Networks](https://mlsanity.com/paper/2107.03257/) (2021)
- [Identifying optimal cycles in quantum thermal machines with reinforcement-learning](https://mlsanity.com/paper/2108.13525/) (2021)

## Built on by (most cited first, 30 of 41)

- [Challenges and Opportunities in Quantum Machine Learning](https://mlsanity.com/paper/2303.09491/) (2023)
- [Barren Plateaus in Variational Quantum Computing](https://mlsanity.com/paper/2405.00781/) (2024)
- [Exponential concentration in quantum kernel methods](https://mlsanity.com/paper/2208.11060/) (2022)
- [Does provable absence of barren plateaus imply classical simulability?](https://mlsanity.com/paper/2312.09121/) (2023)
- [Theory for Equivariant Quantum Neural Networks](https://mlsanity.com/paper/2210.08566/) (2022)
- [Theoretical Guarantees for Permutation-Equivariant Quantum Neural Networks](https://mlsanity.com/paper/2210.09974/) (2022)
- [Understanding quantum machine learning also requires rethinking generalization](https://mlsanity.com/paper/2306.13461/) (2023)
- [Representation Theory for Geometric Quantum Machine Learning](https://mlsanity.com/paper/2210.07980/) (2022)
- [Recent Advances for Quantum Neural Networks in Generative Learning](https://mlsanity.com/paper/2206.03066/) (2022)
- [Quantum Convolutional Neural Networks are Effectively Classically Simulable](https://mlsanity.com/paper/2408.12739/) (2024)
- [VQC-Based Reinforcement Learning with Data Re-uploading: Performance and Trainability](https://mlsanity.com/paper/2401.11555/) (2024)
- [A Primer on Quantum Machine Learning](https://mlsanity.com/paper/2511.15969/) (2025)
- [Generalization of Quantum Machine Learning Models Using Quantum Fisher Information Metric](https://mlsanity.com/paper/2303.13462/) (2023)
- [Splitting and Parallelizing of Quantum Convolutional Neural Networks for Learning Translationally Symmetric Data](https://mlsanity.com/paper/2306.07331/) (2023)
- [Symmetry-invariant quantum machine learning force fields](https://mlsanity.com/paper/2311.11362/) (2023)
- [Trade-off between Gradient Measurement Efficiency and Expressivity in Deep Quantum Neural Networks](https://mlsanity.com/paper/2406.18316/) (2024)
- [Unitary convolutions for learning on graphs and groups](https://mlsanity.com/paper/2410.05499/) (2024)
- [SnCQA: A hardware-efficient equivariant quantum convolutional circuit architecture](https://mlsanity.com/paper/2211.12711/) (2022)
- [On the universality of Sn-equivariant k-body gates](https://mlsanity.com/paper/2303.00728/) (2023)
- [All you need is spin: SU(2) equivariant variational quantum circuits based on spin networks](https://mlsanity.com/paper/2309.07250/) (2023)
- [TensorRL-QAS: Reinforcement learning with tensor networks for improved quantum architecture search](https://mlsanity.com/paper/2505.09371/) (2025)
- [A Comparison Between Invariant and Equivariant Classical and Quantum Graph Neural Networks](https://mlsanity.com/paper/2311.18672/) (2023)
- [mathbbZ2× mathbbZ2 Equivariant Quantum Neural Networks: Benchmarking against Classical Neural Networks](https://mlsanity.com/paper/2311.18744/) (2023)
- [The role of data embedding in equivariant quantum convolutional neural networks](https://mlsanity.com/paper/2312.13250/) (2023)
- [Graph Neural Networks on Quantum Computers](https://mlsanity.com/paper/2405.17060/) (2024)
- [Resource-efficient equivariant quantum convolutional neural networks](https://mlsanity.com/paper/2410.01252/) (2024)
- [Image Classification with Rotation-Invariant Variational Quantum Circuits](https://mlsanity.com/paper/2403.15031/) (2024)
- [Addressing the Current Challenges of Quantum Machine Learning through Multi-Chip Ensembles](https://mlsanity.com/paper/2505.08782/) (2025)
- [Hamiltonian-based Quantum Reinforcement Learning for Neural Combinatorial Optimization](https://mlsanity.com/paper/2405.07790/) (2024)
- [Architectures and random properties of symplectic quantum circuits](https://mlsanity.com/paper/2405.10264/) (2024)

## Abstract (arXiv)

Variational quantum algorithms are the leading candidate for advantage on near-term quantum hardware. When training a parametrized quantum circuit in this setting to solve a specific problem, the choice of ansatz is one of the most important factors that determines the trainability and performance of the algorithm. In quantum machine learning (QML), however, the literature on ansatzes that are motivated by the training data structure is scarce. In this work, we introduce an ansatz for learning tasks on weighted graphs that respects an important graph symmetry, namely equivariance under node permutations. We evaluate the performance of this ansatz on a complex learning task, namely neural combinatorial optimization, where a machine learning model is used to learn a heuristic for a combinatorial optimization problem. We analytically and numerically study the performance of our model, and our results strengthen the notion that symmetry-preserving ansatzes are a key to success in QML.