mlsanityData to 7 Oct 2026

Representation Theory for Geometric Quantum Machine Learning

arXiv 2210.07980 ↗First posted 14 Oct 2022quant-ph cs.LG math.RTcited by 40

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Abstract (arXiv)

Recent advances in classical machine learning have shown that creating models with inductive biases encoding the symmetries of a problem can greatly improve performance. Importation of these ideas, combined with an existing rich body of work at the nexus of quantum theory and symmetry, has given rise to the field of Geometric Quantum Machine Learning (GQML). Following the success of its classical counterpart, it is reasonable to expect that GQML will play a crucial role in developing problem-specific and quantum-aware models capable of achieving a computational advantage. Despite the simplicity of the main idea of GQML -- create architectures respecting the symmetries of the data -- its practical implementation requires a significant amount of knowledge of group representation theory. We present an introduction to representation theory tools from the optics of quantum learning, driven by key examples involving discrete and continuous groups. These examples are sewn together by an exposition outlining the formal capture of GQML symmetries via "label invariance under the action of a group representation", a brief (but rigorous) tour through finite and compact Lie group representation theory, a reexamination of ubiquitous tools like Haar integration and twirling, and an overview of some successful strategies for detecting symmetries.

Key facts. Cited by 40 papers in the map (data through 7 Oct 2026). Builds on 38 of its 146 references that are in the map. First posted on arXiv 14 Oct 2022.

Builds on

38 of 146 references.

References · 38

  1. 2016Group Equivariant Convolutional Networkscited by 1,154
  2. 2021Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gaugescited by 1,131
  3. 2020Variational Quantum Algorithmscited by 687
  4. 2018Supervised learning with quantum enhanced feature spacescited by 654
  5. 2018Barren plateaus in quantum neural network training landscapescited by 624
  6. 2020The power of quantum neural networkscited by 377
  7. 2018Invariant and Equivariant Graph Networkscited by 339
  8. 2018On the Generalization of Equivariance and Convolution in Neural Networks to the Action of Compact Groupscited by 296
  9. 2020A rigorous and robust quantum speed-up in supervised machine learningcited by 270
  10. 2019Gauge Equivariant Convolutional Networks and the Icosahedral CNNcited by 262
  11. 2019Cormorant: Covariant Molecular Neural Networkscited by 262
  12. 2020Predicting Many Properties of a Quantum System from Very Few Measurementscited by 229
26 more
  1. 2018A General Theory of Equivariant CNNs on Homogeneous Spacescited by 223
  2. 2021Generalization in quantum machine learning from few training datacited by 214
  3. 2021Quantum advantage in learning from experimentscited by 186
  4. 2021Connecting ansatz expressibility to gradient magnitudes and barren plateauscited by 180
  5. 2018Clebsch-Gordan Nets: a Fully Fourier Space Spherical Convolutional Neural Networkcited by 168
  6. 2020Absence of Barren Plateaus in Quantum Convolutional Neural Networkscited by 124
  7. 2020Entanglement Induced Barren Plateauscited by 101
  8. 2021Theory of overparametrization in quantum neural networkscited by 100
  9. 2022Approximately Equivariant Networks for Imperfectly Symmetric Dynamicscited by 95
  10. 2018Roto-Translation Covariant Convolutional Networks for Medical Image Analysiscited by 91
  11. 2021Provably efficient machine learning for quantum many-body problemscited by 91
  12. 2021Provably Strict Generalisation Benefit for Equivariant Modelscited by 90
  13. 2022Exploiting symmetry in variational quantum machine learningcited by 68
  14. 2019Quantum Graph Neural Networkscited by 66
  15. 2020Trainability of Dissipative Perceptron-Based Quantum Neural Networkscited by 56
  16. 2022Group-Invariant Quantum Machine Learningcited by 56
  17. 2021Equivalence of quantum barren plateaus to cost concentration and narrow gorgescited by 53
  18. 2021Exponential separations between learning with and without quantum memorycited by 50
  19. 2022Equivariant quantum circuits for learning on weighted graphscited by 41
  20. 2021Subtleties in the trainability of quantum machine learning modelscited by 33
  21. 2021Speeding up Learning Quantum States through Group Equivariant Convolutional Quantum Ans"atzecited by 29
  22. 2020Reformulation of the No-Free-Lunch Theorem for Entangled Data Setscited by 28
  23. 2021Entangled Datasets for Quantum Machine Learningcited by 26
  24. 2021Quantum algorithms for group convolution, cross-correlation, and equivariant transformationscited by 5
  25. 2022Linear programming with unitary-equivariant constraintscited by 3
  26. 2021Testing symmetry on quantum computerscited by 2

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2022: 4; 2023: 15; 2024: 16; 2025: 4; 2026: 1.

40 papers.

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  1. 2024Barren Plateaus in Variational Quantum Computingcited by 130
  2. 2023Does provable absence of barren plateaus imply classical simulability?cited by 78
  3. 2022Theory for Equivariant Quantum Neural Networkscited by 60
  4. 2022Theoretical Guarantees for Permutation-Equivariant Quantum Neural Networkscited by 52
  5. 2023Trainability barriers and opportunities in quantum generative modelingcited by 35
  6. 2024Quantum Convolutional Neural Networks are Effectively Classically Simulablecited by 33
  7. 2022Problem-Dependent Power of Quantum Neural Networks on Multi-Class Classificationcited by 20
  8. 2024On the hardness of learning under symmetriescited by 13
  9. 2023Statistical Complexity of Quantum Learningcited by 11
  10. 2023Generalization of Quantum Machine Learning Models Using Quantum Fisher Information Metriccited by 10
  11. 2023Splitting and Parallelizing of Quantum Convolutional Neural Networks for Learning Translationally Symmetric Datacited by 10
  12. 2023Approximately Equivariant Quantum Neural Network for p4m Group Symmetries in Imagescited by 10
28 more
  1. 2025A Primer on Quantum Machine Learningcited by 9
  2. 2023Analyzing Convergence in Quantum Neural Networks: Deviations from Neural Tangent Kernelscited by 8
  3. 2024Reinforcement learning-based architecture search for quantum machine learningcited by 8
  4. 2022SnCQA: A hardware-efficient equivariant quantum convolutional circuit architecturecited by 7
  5. 2023On the universality of Sn-equivariant k-body gatescited by 7
  6. 2023All you need is spin: SU(2) equivariant variational quantum circuits based on spin networkscited by 7
  7. 2024Trade-off between Gradient Measurement Efficiency and Expressivity in Deep Quantum Neural Networkscited by 7
  8. 2023Time-Warping Invariant Quantum Recurrent Neural Networks via Quantum-Classical Adaptive Gatingcited by 5
  9. 20233D-QAE: Fully Quantum Auto-Encoding of 3D Point Cloudscited by 5
  10. 2024Learning with SASQuaTCh: a Novel Variational Quantum Transformer Architecture with Kernel-Based Self-Attentioncited by 5
  11. 2023The role of data embedding in equivariant quantum convolutional neural networkscited by 4
  12. 2024Graph Neural Networks on Quantum Computerscited by 4
  13. 2024Resource-efficient equivariant quantum convolutional neural networkscited by 4
  14. 2023Quantum Machine Learning for Malware Classificationcited by 3
  15. 2024Image Classification with Rotation-Invariant Variational Quantum Circuitscited by 3
  16. 2024Prospects of Privacy Advantage in Quantum Machine Learningcited by 3
  17. 2025Enhancing variational quantum algorithms by balancing training on classical and quantum hardwarecited by 3
  18. 2024Barren plateaus are amplified by the dimension of quditscited by 2
  19. 2026A PAC-Bayesian approach to generalization for quantum modelscited by 2
  20. 2024Architectures and random properties of symplectic quantum circuitscited by 1
  21. 2023A quantum tug of war between randomness and symmetries on homogeneous spacescited by 0
  22. 2023Neural Crystalscited by 0
  23. 2024Permutation-equivariant quantum convolutional neural networkscited by 0
  24. 2024Reverse Map Projections as Equivariant Quantum Embeddingscited by 0
  25. 2024Can Geometric Quantum Machine Learning Lead to Advantage in Barcode Classification?cited by 0
  26. 2024Exploring Channel Distinguishability in Local Neighborhoods of the Model Space in Quantum Neural Networkscited by 0
  27. 2025K-P Quantum Neural Networkscited by 0
  28. 2025PERM EQ x GRAPH EQ: Equivariant Neural Networks for Quantum Molecular Learningcited by 0