mlsanityData to 14 Oct 2026

Speeding up Learning Quantum States through Group Equivariant Convolutional Quantum Ans"atze

arXiv 2112.07611 ↗First posted 14 Dec 2021quant-ph cs.AI cs.LGcited by 29

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Abstract (arXiv)

We develop a theoretical framework for Sn-equivariant convolutional quantum circuits with SU(d)-symmetry, building on and significantly generalizing Jordan's Permutational Quantum Computing (PQC) formalism based on Schur-Weyl duality connecting both SU(d) and Sn actions on qudits. In particular, we utilize the Okounkov-Vershik approach to prove Harrow's statement (Ph.D. Thesis 2005 p.160) on the equivalence between operatornameSU(d) and Sn irrep bases and to establish the Sn-equivariant Convolutional Quantum Alternating Ans"atze (Sn-CQA) using Young-Jucys-Murphy (YJM) elements. We prove that Sn-CQA is able to generate any unitary in any given Sn irrep sector, which may serve as a universal model for a wide array of quantum machine learning problems with the presence of SU(d) symmetry. Our method provides another way to prove the universality of Quantum Approximate Optimization Algorithm (QAOA) and verifies that 4-local SU(d) symmetric unitaries are sufficient to build generic SU(d) symmetric quantum circuits up to relative phase factors. We present numerical simulations to showcase the effectiveness of the ans"atze to find the ground state energy of the J1--J2 antiferromagnetic Heisenberg model on the rectangular and Kagome lattices. Our work provides the first application of the celebrated Okounkov-Vershik's Sn representation theory to quantum physics and machine learning, from which to propose quantum variational ans"atze that strongly suggests to be classically intractable tailored towards a specific optimization problem.

Key facts. Cited by 29 papers in the map (data through 14 Oct 2026). Builds on 13 of its 85 references that are in the map. First posted on arXiv 14 Dec 2021.

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13 of 85 references.

References · 13

  1. 2020Variational Quantum Algorithmscited by 692
  2. 2018Supervised learning with quantum enhanced feature spacescited by 665
  3. 2018Barren plateaus in quantum neural network training landscapescited by 630
  4. 2016Quantum Machine Learningcited by 326
  5. 2018On the Generalization of Equivariance and Convolution in Neural Networks to the Action of Compact Groupscited by 300
  6. 2020A rigorous and robust quantum speed-up in supervised machine learningcited by 271
  7. 2021Supervised quantum machine learning models are kernel methodscited by 200
  8. 2018Clebsch-Gordan Nets: a Fully Fourier Space Spherical Convolutional Neural Networkcited by 171
  9. 2020Lorentz Group Equivariant Neural Network for Particle Physicscited by 121
  10. 2021Information-theoretic bounds on quantum advantage in machine learningcited by 96
  11. 2020The general theory of permutation equivarant neural networks and higher order graph variational encoderscited by 28
  12. 2020Neural network wave functions and the sign problemcited by 20
1 more
  1. 2021Learning ground states of quantum Hamiltonians with graph networkscited by 12

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Citing papers per year: 2022: 12; 2023: 9; 2024: 7; 2025: 1.

29 papers.

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  1. 2023Does provable absence of barren plateaus imply classical simulability?cited by 80
  2. 2022Theory for Equivariant Quantum Neural Networkscited by 62
  3. 2022Group-Invariant Quantum Machine Learningcited by 56
  4. 2022Theoretical Guarantees for Permutation-Equivariant Quantum Neural Networkscited by 52
  5. 2022Representation Theory for Geometric Quantum Machine Learningcited by 40
  6. 2022Symmetric Pruning in Quantum Neural Networkscited by 21
  7. 2023Generalization of Quantum Machine Learning Models Using Quantum Fisher Information Metriccited by 10
  8. 2023Splitting and Parallelizing of Quantum Convolutional Neural Networks for Learning Translationally Symmetric Datacited by 10
  9. 2023Approximately Equivariant Quantum Neural Network for p4m Group Symmetries in Imagescited by 10
  10. 2023Symmetry-invariant quantum machine learning force fieldscited by 8
  11. 2024Trade-off between Gradient Measurement Efficiency and Expressivity in Deep Quantum Neural Networkscited by 8
  12. 2022Lattice Convolutional Networks for Learning Ground States of Quantum Many-Body Systemscited by 7
17 more
  1. 2022SnCQA: A hardware-efficient equivariant quantum convolutional circuit architecturecited by 7
  2. 2023On the universality of Sn-equivariant k-body gatescited by 7
  3. 2023All you need is spin: SU(2) equivariant variational quantum circuits based on spin networkscited by 7
  4. 2022Toward Super-polynomial Quantum Speedup of Equivariant Quantum Algorithms with SU(d) Symmetrycited by 4
  5. 2023The role of data embedding in equivariant quantum convolutional neural networkscited by 4
  6. 2024Graph Neural Networks on Quantum Computerscited by 4
  7. 2024Resource-efficient equivariant quantum convolutional neural networkscited by 4
  8. 2022Efficient Quantum Feature Extraction for CNN-based Learningcited by 3
  9. 2022An Empirical Review of Optimization Techniques for Quantum Variational Circuitscited by 3
  10. 2022Learning quantum symmetries with interactive quantum-classical variational algorithmscited by 1
  11. 2022Active Learning on a Programmable Photonic Quantum Processorcited by 1
  12. 2024Architectures and random properties of symplectic quantum circuitscited by 1
  13. 2024Probing many-body Bell correlation depth with superconducting qubitscited by 1
  14. 2023A quantum tug of war between randomness and symmetries on homogeneous spacescited by 0
  15. 2024Permutation-equivariant quantum convolutional neural networkscited by 0
  16. 2024Quantum-machine-assisted Drug Discoverycited by 0
  17. 2025PERM EQ x GRAPH EQ: Equivariant Neural Networks for Quantum Molecular Learningcited by 0