# Theoretical Guarantees for Permutation-Equivariant Quantum Neural Networks

arXiv: https://arxiv.org/abs/2210.09974 · HTML: https://mlsanity.com/paper/2210.09974/

## Key facts

- Cited by 52 papers in the map (data through 14 Oct 2026).
- Builds on 67 of its 173 references that are in the map.
- First posted on arXiv 18 Oct 2022.

## Builds on (most cited first, 30 of 67)

- [Deep Sets](https://mlsanity.com/paper/1703.06114/) (2017)
- [Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges](https://mlsanity.com/paper/2104.13478/) (2021)
- [Group Equivariant Convolutional Networks](https://mlsanity.com/paper/1602.07576/) (2016)
- [A Convergence Theory for Deep Learning via Over-Parameterization](https://mlsanity.com/paper/1811.03962/) (2018)
- [Tensor field networks: Rotation- and translation-equivariant neural networks for 3D point clouds](https://mlsanity.com/paper/1802.08219/) (2018)
- [Variational Quantum Algorithms](https://mlsanity.com/paper/2012.09265/) (2020)
- [Supervised learning with quantum enhanced feature spaces](https://mlsanity.com/paper/1804.11326/) (2018)
- [Gradient Descent Provably Optimizes Over-parameterized Neural Networks](https://mlsanity.com/paper/1810.02054/) (2018)
- [Barren plateaus in quantum neural network training landscapes](https://mlsanity.com/paper/1803.11173/) (2018)
- [SchNet: A continuous-filter convolutional neural network for modeling quantum interactions](https://mlsanity.com/paper/1706.08566/) (2017)
- [Provably Powerful Graph Networks](https://mlsanity.com/paper/1905.11136/) (2019)
- [The power of quantum neural networks](https://mlsanity.com/paper/2011.00027/) (2020)
- [Invariant and Equivariant Graph Networks](https://mlsanity.com/paper/1812.09902/) (2018)
- [Learning and Generalization in Overparameterized Neural Networks, Going Beyond Two Layers](https://mlsanity.com/paper/1811.04918/) (2018)
- [Power of data in quantum machine learning](https://mlsanity.com/paper/2011.01938/) (2020)
- [Quantum Machine Learning](https://mlsanity.com/paper/1611.09347/) (2016)
- [On the Generalization of Equivariance and Convolution in Neural Networks to the Action of Compact Groups](https://mlsanity.com/paper/1802.03690/) (2018)
- [Cormorant: Covariant Molecular Neural Networks](https://mlsanity.com/paper/1906.04015/) (2019)
- [Equivariant Flows: Exact Likelihood Generative Learning for Symmetric Densities](https://mlsanity.com/paper/2006.02425/) (2020)
- [Deep learning: a statistical viewpoint](https://mlsanity.com/paper/2103.09177/) (2021)
- [Generalization in quantum machine learning from few training data](https://mlsanity.com/paper/2111.05292/) (2021)
- [Noise-Induced Barren Plateaus in Variational Quantum Algorithms](https://mlsanity.com/paper/2007.14384/) (2020)
- [Universal Invariant and Equivariant Graph Neural Networks](https://mlsanity.com/paper/1905.04943/) (2019)
- [Connecting ansatz expressibility to gradient magnitudes and barren plateaus](https://mlsanity.com/paper/2101.02138/) (2021)
- [On the Universality of Invariant Networks](https://mlsanity.com/paper/1901.09342/) (2019)
- [Hamiltonian Generative Networks](https://mlsanity.com/paper/1909.13789/) (2019)
- [Quantum generative adversarial networks](https://mlsanity.com/paper/1804.08641/) (2018)
- [Absence of Barren Plateaus in Quantum Convolutional Neural Networks](https://mlsanity.com/paper/2011.02966/) (2020)
- [Lorentz Group Equivariant Neural Network for Particle Physics](https://mlsanity.com/paper/2006.04780/) (2020)
- [Entanglement Induced Barren Plateaus](https://mlsanity.com/paper/2010.15968/) (2020)

## Built on by (most cited first, 30 of 52)

- [Barren Plateaus in Variational Quantum Computing](https://mlsanity.com/paper/2405.00781/) (2024)
- [Does provable absence of barren plateaus imply classical simulability?](https://mlsanity.com/paper/2312.09121/) (2023)
- [Understanding quantum machine learning also requires rethinking generalization](https://mlsanity.com/paper/2306.13461/) (2023)
- [Quantum Convolutional Neural Networks are Effectively Classically Simulable](https://mlsanity.com/paper/2408.12739/) (2024)
- [Trainability barriers and opportunities in quantum generative modeling](https://mlsanity.com/paper/2305.02881/) (2023)
- [On the relation between trainability and dequantization of variational quantum learning models](https://mlsanity.com/paper/2406.07072/) (2024)
- [Problem-Dependent Power of Quantum Neural Networks on Multi-Class Classification](https://mlsanity.com/paper/2301.01597/) (2022)
- [On the hardness of learning under symmetries](https://mlsanity.com/paper/2401.01869/) (2024)
- [The power and limitations of learning quantum dynamics incoherently](https://mlsanity.com/paper/2303.12834/) (2023)
- [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)
- [Approximately Equivariant Quantum Neural Network for p4m Group Symmetries in Images](https://mlsanity.com/paper/2310.02323/) (2023)
- [Effects of noise on the overparametrization of quantum neural networks](https://mlsanity.com/paper/2302.05059/) (2023)
- [Adversarial Robustness Guarantees for Quantum Classifiers](https://mlsanity.com/paper/2405.10360/) (2024)
- [Reinforcement learning-based architecture search for quantum machine learning](https://mlsanity.com/paper/2406.02717/) (2024)
- [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)
- [A Comparison Between Invariant and Equivariant Classical and Quantum Graph Neural Networks](https://mlsanity.com/paper/2311.18672/) (2023)
- [Time-Warping Invariant Quantum Recurrent Neural Networks via Quantum-Classical Adaptive Gating](https://mlsanity.com/paper/2301.08173/) (2023)
- [3D-QAE: Fully Quantum Auto-Encoding of 3D Point Clouds](https://mlsanity.com/paper/2311.05604/) (2023)
- [Symmetry breaking in geometric quantum machine learning in the presence of noise](https://mlsanity.com/paper/2401.10293/) (2024)
- [The role of data embedding in equivariant quantum convolutional neural networks](https://mlsanity.com/paper/2312.13250/) (2023)
- [MG-Net: Learn to Customize QAOA with Circuit Depth Awareness](https://mlsanity.com/paper/2409.18692/) (2024)
- [Resource-efficient equivariant quantum convolutional neural networks](https://mlsanity.com/paper/2410.01252/) (2024)

## Abstract (arXiv)

Despite the great promise of quantum machine learning models, there are several challenges one must overcome before unlocking their full potential. For instance, models based on quantum neural networks (QNNs) can suffer from excessive local minima and barren plateaus in their training landscapes. Recently, the nascent field of geometric quantum machine learning (GQML) has emerged as a potential solution to some of those issues. The key insight of GQML is that one should design architectures, such as equivariant QNNs, encoding the symmetries of the problem at hand. Here, we focus on problems with permutation symmetry (i.e., the group of symmetry Sn), and show how to build Sn-equivariant QNNs. We provide an analytical study of their performance, proving that they do not suffer from barren plateaus, quickly reach overparametrization, and generalize well from small amounts of data. To verify our results, we perform numerical simulations for a graph state classification task. Our work provides the first theoretical guarantees for equivariant QNNs, thus indicating the extreme power and potential of GQML.