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

arXiv: https://arxiv.org/abs/2112.07611 · HTML: https://mlsanity.com/paper/2112.07611/

## 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.

## Builds on (most cited first, 13 of 13)

- [Variational Quantum Algorithms](https://mlsanity.com/paper/2012.09265/) (2020)
- [Supervised learning with quantum enhanced feature spaces](https://mlsanity.com/paper/1804.11326/) (2018)
- [Barren plateaus in quantum neural network training landscapes](https://mlsanity.com/paper/1803.11173/) (2018)
- [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)
- [A rigorous and robust quantum speed-up in supervised machine learning](https://mlsanity.com/paper/2010.02174/) (2020)
- [Supervised quantum machine learning models are kernel methods](https://mlsanity.com/paper/2101.11020/) (2021)
- [Clebsch-Gordan Nets: a Fully Fourier Space Spherical Convolutional Neural Network](https://mlsanity.com/paper/1806.09231/) (2018)
- [Lorentz Group Equivariant Neural Network for Particle Physics](https://mlsanity.com/paper/2006.04780/) (2020)
- [Information-theoretic bounds on quantum advantage in machine learning](https://mlsanity.com/paper/2101.02464/) (2021)
- [The general theory of permutation equivarant neural networks and higher order graph variational encoders](https://mlsanity.com/paper/2004.03990/) (2020)
- [Neural network wave functions and the sign problem](https://mlsanity.com/paper/2002.04613/) (2020)
- [Learning ground states of quantum Hamiltonians with graph networks](https://mlsanity.com/paper/2110.06390/) (2021)

## Built on by (most cited first, 29 of 29)

- [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)
- [Group-Invariant Quantum Machine Learning](https://mlsanity.com/paper/2205.02261/) (2022)
- [Theoretical Guarantees for Permutation-Equivariant Quantum Neural Networks](https://mlsanity.com/paper/2210.09974/) (2022)
- [Representation Theory for Geometric Quantum Machine Learning](https://mlsanity.com/paper/2210.07980/) (2022)
- [Symmetric Pruning in Quantum Neural Networks](https://mlsanity.com/paper/2208.14057/) (2022)
- [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)
- [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)
- [Lattice Convolutional Networks for Learning Ground States of Quantum Many-Body Systems](https://mlsanity.com/paper/2206.07370/) (2022)
- [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)
- [Toward Super-polynomial Quantum Speedup of Equivariant Quantum Algorithms with SU(d) Symmetry](https://mlsanity.com/paper/2207.07250/) (2022)
- [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)
- [Efficient Quantum Feature Extraction for CNN-based Learning](https://mlsanity.com/paper/2201.01246/) (2022)
- [An Empirical Review of Optimization Techniques for Quantum Variational Circuits](https://mlsanity.com/paper/2202.01389/) (2022)
- [Learning quantum symmetries with interactive quantum-classical variational algorithms](https://mlsanity.com/paper/2206.11970/) (2022)
- [Active Learning on a Programmable Photonic Quantum Processor](https://mlsanity.com/paper/2208.02104/) (2022)
- [Architectures and random properties of symplectic quantum circuits](https://mlsanity.com/paper/2405.10264/) (2024)
- [Probing many-body Bell correlation depth with superconducting qubits](https://mlsanity.com/paper/2406.17841/) (2024)
- [A quantum tug of war between randomness and symmetries on homogeneous spaces](https://mlsanity.com/paper/2309.05253/) (2023)
- [Permutation-equivariant quantum convolutional neural networks](https://mlsanity.com/paper/2404.18198/) (2024)
- [Quantum-machine-assisted Drug Discovery](https://mlsanity.com/paper/2408.13479/) (2024)
- [PERM EQ x GRAPH EQ: Equivariant Neural Networks for Quantum Molecular Learning](https://mlsanity.com/paper/2512.05475/) (2025)

## 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.