Permutation-equivariant quantum convolutional neural networks

Abstract

The Symmetric group $S_n$ manifests itself in large classes of quantum systems as the invariance of certain characteristics of a quantum state with respect to permuting the qubits. The subgroups of $S_n$ arise, among many other contexts, to describe label symmetry of classical images with respect to spatial transformations, e.g. reflection or rotation. Equipped with the formalism of geometric quantum machine learning, in this work we propose the architectures of equivariant quantum convolutional neural networks (EQCNNs) adherent to $S_n$ and its subgroups. We demonstrate that a careful choice of pixel-to-qubit embedding order can facilitate easy construction of EQCNNs for small subgroups of $S_n$. Our novel EQCNN architecture corresponding to the full permutation group $S_n$ is built by applying all possible QCNNs with equal probability, which can also be conceptualized as a dropout strategy in quantum neural networks. For subgroups of $S_n$, our numerical results using MNIST datasets show better classification accuracy than non-equivariant QCNNs. The $S_n$-equivariant QCNN architecture shows significantly improved training and test performance than non-equivariant QCNN for classification of connected and non-connected graphs. When trained with sufficiently large number of data, the $S_n$-equivariant QCNN shows better average performance compared to $S_n$-equivariant QNN . These results contribute towards building powerful quantum machine learning architectures in permutation-symmetric systems.

13 pages, 10 figures

References in corpus (50)

• Quantum Machine Learning
• A variational eigenvalue solver on a quantum processor
• Barren plateaus in quantum neural network training landscapes
• The theory of variational hybrid quantum-classical algorithms
• Quantum support vector machine for big data classification
• Measurement-based quantum computation with cluster states
• Quantum Circuit Learning
• Quantum Convolutional Neural Networks
• An introduction to quantum machine learning
• Parameterized quantum circuits as machine learning models
• Cost Function Dependent Barren Plateaus in Shallow Parametrized Quantum Circuits
• Multi-party entanglement in graph states
• Circuit-centric quantum classifiers
• Expressibility and entangling capability of parameterized quantum circuits for hybrid quantum-classical algorithms
• A class of quantum many-body states that can be efficiently simulated
• Connecting ansatz expressibility to gradient magnitudes and barren plateaus
• Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges
• Quantum advantage in learning from experiments
• A rigorous and robust quantum speed-up in supervised machine learning
• Quantum autoencoders for efficient compression of quantum data
• Generalization in quantum machine learning from few training data
• Quantum convolutional neural network for classical data classification
• Opportunities and challenges for quantum-assisted machine learning in near-term quantum computers
• Is quantum advantage the right goal for quantum machine learning?
• Exploiting symmetry in variational quantum machine learning
• Group-Invariant Quantum Machine Learning
• Realizing Quantum Convolutional Neural Networks on a Superconducting Quantum Processor to Recognize Quantum Phases
• Hybrid Quantum-Classical Generative Adversarial Network for High Resolution Image Generation
• Theory for Equivariant Quantum Neural Networks
• Theoretical Guarantees for Permutation-Equivariant Quantum Neural Networks
• A Universal Training Algorithm for Quantum Deep Learning
• Generative quantum machine learning via denoising diffusion probabilistic models
• Quantum phase detection generalisation from marginal quantum neural network models
• Speeding up Learning Quantum States through Group Equivariant Convolutional Quantum Ansätze
• Reflection Equivariant Quantum Neural Networks for Enhanced Image Classification
• BrainIB: Interpretable Brain Network-based Psychiatric Diagnosis with Graph Information Bottleneck
• Provably Trainable Rotationally Equivariant Quantum Machine Learning
• Model-Independent Learning of Quantum Phases of Matter with Quantum Convolutional Neural Networks
• Overfitting in quantum machine learning and entangling dropout
• Quantum-Noise-Driven Generative Diffusion Models
• Representation Theory for Geometric Quantum Machine Learning
• An efficient quantum algorithm for preparation of uniform quantum superposition states
• A General Approach to Dropout in Quantum Neural Networks
• Approximately Equivariant Quantum Neural Network for $p4m$ Group Symmetries in Images
• How to enhance quantum generative adversarial learning of noisy information
• The role of data embedding in equivariant quantum convolutional neural networks
• Splitting and Parallelizing of Quantum Convolutional Neural Networks for Learning Translationally Symmetric Data
• Equivariant Quantum Graph Circuits
• Toward Super-polynomial Quantum Speedup of Equivariant Quantum Algorithms with SU($d$) Symmetry
• All you need is spin: SU(2) equivariant variational quantum circuits based on spin networks

Cited by in corpus (1)

• The role of data embedding in equivariant quantum convolutional neural networks