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Symmetry breaking in geometric quantum machine learning in the presence of noise

Cenk Tüysüz, Su Yeon Chang, Maria Demidik, Karl Jansen, Sofia Vallecorsa, Michele Grossi

TL;DR

Geometric quantum machine learning with equivariant neural networks is analyzed under hardware noise to understand symmetry preservation. The authors show that realistic Pauli channels can preserve equivariance, while amplitude damping introduces symmetry breaking that scales roughly as $d\gamma$ with circuit depth $d$ and noise strength $\gamma$, a finding validated by simulations and hardware up to 64 qubits. They introduce practical metrics (generalized McNemar-Bowker test and Label Misassignment) and mitigation strategies (adaptive thresholding, $Z^{\otimes n}$ representation, and pulse-efficient transpilation) to bolster symmetry protection on noisy devices. These results guide the robust deployment of EQNNs on NISQ hardware and motivate further exploration of symmetry-aware error mitigation and scalable geometric quantum learning.

Abstract

Geometric quantum machine learning based on equivariant quantum neural networks (EQNN) recently appeared as a promising direction in quantum machine learning. Despite the encouraging progress, the studies are still limited to theory, and the role of hardware noise in EQNN training has never been explored. This work studies the behavior of EQNN models in the presence of noise. We show that certain EQNN models can preserve equivariance under Pauli channels, while this is not possible under the amplitude damping channel. We claim that the symmetry breaking grows linearly in the number of layers and noise strength. We support our claims with numerical data from simulations as well as hardware up to 64 qubits. Furthermore, we provide strategies to enhance the symmetry protection of EQNN models in the presence of noise.

Symmetry breaking in geometric quantum machine learning in the presence of noise

TL;DR

Geometric quantum machine learning with equivariant neural networks is analyzed under hardware noise to understand symmetry preservation. The authors show that realistic Pauli channels can preserve equivariance, while amplitude damping introduces symmetry breaking that scales roughly as with circuit depth and noise strength , a finding validated by simulations and hardware up to 64 qubits. They introduce practical metrics (generalized McNemar-Bowker test and Label Misassignment) and mitigation strategies (adaptive thresholding, representation, and pulse-efficient transpilation) to bolster symmetry protection on noisy devices. These results guide the robust deployment of EQNNs on NISQ hardware and motivate further exploration of symmetry-aware error mitigation and scalable geometric quantum learning.

Abstract

Geometric quantum machine learning based on equivariant quantum neural networks (EQNN) recently appeared as a promising direction in quantum machine learning. Despite the encouraging progress, the studies are still limited to theory, and the role of hardware noise in EQNN training has never been explored. This work studies the behavior of EQNN models in the presence of noise. We show that certain EQNN models can preserve equivariance under Pauli channels, while this is not possible under the amplitude damping channel. We claim that the symmetry breaking grows linearly in the number of layers and noise strength. We support our claims with numerical data from simulations as well as hardware up to 64 qubits. Furthermore, we provide strategies to enhance the symmetry protection of EQNN models in the presence of noise.
Paper Structure (23 sections, 1 theorem, 45 equations, 17 figures, 1 table)

This paper contains 23 sections, 1 theorem, 45 equations, 17 figures, 1 table.

Table of Contents

  1. Introduction
  2. Framework
  3. Equivariant Quantum Neural Networks
  4. Noise models
  5. Concentration of measure
  6. Equivariance under noise
  7. Toy model
  8. Quantifying symmetry breaking
  9. Experiments
  10. Binary classification
  11. Symmetry breaking
  12. Two qubit case
  13. Multi-qubit case
  14. Conclusion
  15. Noise models
  16. ...and 8 more sections

Key Result

Lemma 1

A quantum learning model which consists of equivariant embedding, equivariant quantum circuit ansatz and invariant observable with respect to a symmetry group $\mathcal{S}$ is invariant with respect to $\mathcal{S}$ :

Figures (17)

  • Figure 1: Drawing of the local noise model. A circuit with input $\rho$ and layers $U_i$, where the local $\Lambda$ representing the action of noise are applied after each layer.
  • Figure 2: One qubit toy model under noise with identity gates decomposed into unitaries $U$ and $U^\dagger$, $d$ times, i.e., $I = (UU^\dagger)^d$.
  • Figure 3: An ad-hoc dataset with $\mathcal{Z}_2$ label symmetry such that $R(\sigma)\cdot(\mathbf{x}_i)=-\mathbf{x}_i$.
  • Figure 4: Two qubit circuits used in the experiments.
  • Figure 5: Binary classification results under noise channels. All models are trained with ten different initializations and layers varied from 1-10. The test accuracy, averaged over the runs, is plotted for the best-performing layer of the corresponding model. Noise strength $p$ in the case of the DP channel and $\gamma$ in the case of the AD channel is varied from 0.0 to 0.1 with 0.01 increments. a) Results under DP channel. b) Results under AD channel. c) Results under AD channel with and without using adaptive thresholding (AT) during training.
  • ...and 12 more figures

Theorems & Definitions (7)

  • Definition 1: Invariance
  • Definition 2: Equivariant Embedding
  • Definition 3: Equivariant Gate
  • Lemma 1: Invariance from equivariance
  • Definition 4: Exponential concentration
  • Definition 5: $\mathcal{Z}_2$ generalized MB test
  • Definition 6: Label Misassignment (LM)