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arXiv:2404.18198 · Quantum Science and Technology

Permutation-equivariant quantum convolutional neural networks

Sreetama Das, Filippo Caruso

Published Nov 4, 2024• 4 citations
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Abstract

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. Subgroups of S n arise, among many other contexts, to describe label symmetry of classical images with respect to spatial transformations, such as reflection or rotation. Equipped with the formalism of geometric quantum machine learning, in this study 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.

DOI
10.1088/2058-9565/ad8e80

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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. Subgroups of S n arise, among many other contexts, to describe label symmetry of classical images with respect to spatial transformations, such as reflection or rotation. Equipped with the formalism of geometric quantum machine learning, in this study 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.
Abstract Geometric deep learning refers to the scenario in which the symmetries of a dataset are used to constrain the parameter space of a neural network and thus, improve their trainability and generalization. Recently, this idea has been incorporated into the field of quantum machine learning, which has given rise to equivariant quantum neural networks (EQNNs). In this work, we investigate the role of classical-to-quantum embedding on the performance of equivariant quantum convolutional neural networks (EQCNNs) for the classification of images. We discuss the connection between the data embedding method and the resulting representation of a symmetry group and analyze how changing representation affects the expressibility of an EQCNN. We numerically compare the classification accuracy of EQCNNs with three different basis-permuted amplitude embeddings to the one obtained from a non-equivariant quantum convolutional neural network (QCNN). Our results show a clear dependence of classification accuracy on the underlying embedding, especially for initial training iterations. The improvement in classification accuracy of EQCNN over non-equivariant QCNN may be present or absent depending on the particular embedding and dataset used. The noisy simulation using simple noise models shows that certain EQCNNs are more robust to noise than non-equivariant QCNNs. It is expected that the results of this work can be useful to the community for a better understanding of the importance of data embedding choice in the context of geometric quantum machine learning.
Quantum Neural Networks (QNNs) are suggested as one of the quantum algorithms which can be efficiently simulated with a low depth on near-term quantum hardware in the presence of noises. However, their performance highly relies on choosing the most suitable architecture of Variational Quantum Algorithms (VQAs), and the problem-agnostic models often suffer issues regarding trainability and generalization power. As a solution, the most recent works explore Geometric Quantum Machine Learning (GQML) using QNNs equivariant with respect to the underlying symmetry of the dataset. GQML adds an inductive bias to the model by incorporating the prior knowledge on the given dataset and leads to enhancing the optimization performance while constraining the search space. This work proposes equivariant Quantum Convolutional Neural Networks (EquivQCNNs) for image classification under planar p4m symmetry, including reflectional and 90° rotational symmetry. We present the results tested in different use cases, such as phase detection of the 2D Ising model and classification of the extended MNIST dataset, and compare them with those obtained with the non-equivariant model, proving that the equivariance fosters better generalization of the model.
Abstract Despite the great promise of quantum machine learning models, there are several challenges one must overcome before unlocking their full potential. For instance, models based on quantum neural networks (QNNs) can suffer from excessive local minima and barren plateaus in their training landscapes. Recently, the nascent field of geometric quantum machine learning (GQML) has emerged as a potential solution to some of those issues. The key insight of GQML is that one should design architectures, such as equivariant QNNs, encoding the symmetries of the problem at hand. Here, we focus on problems with permutation symmetry (i.e., symmetry group Sn), and show how to build Sn-equivariant QNNs We provide an analytical study of their performance, proving that they do not suffer from barren plateaus, quickly reach overparametrization, and generalize well from small amounts of data. To verify our results, we perform numerical simulations for a graph state classification task. Our work provides theoretical guarantees for equivariant QNNs, thus indicating the power and potential of GQML.
Abstract Equivariant quantum neural networks (QNNs) are promising variational models that exploit symmetries to improve machine learning capabilities. Despite theoretical developments in equivariant QNNs, their implementation on near-term quantum devices remains challenging due to limited computational resources. This study proposes a resource-efficient model of equivariant quantum convolutional networks (QCNNs) called equivariant split-parallelizing QCNN (sp-QCNN). Using a group-theoretical approach, we encode general symmetries into our model beyond the translational symmetry addressed by previous sp-QCNNs. We achieve this by splitting the circuit at the pooling layer while preserving symmetry. This splitting structure effectively parallelizes QCNNs to improve measurement efficiency in estimating the expectation value of an observable and its gradient by order of the number of qubits. Our model also exhibits high trainability and generalization performance, including the absence of barren plateaus. Numerical experiments demonstrate that the equivariant sp-QCNN can be trained and generalized with fewer measurement resources than a conventional equivariant QCNN in a noisy quantum data classification task. Our results contribute to the advancement of practical quantum machine learning algorithms.
Equivariant quantum graph neural networks (EQGNNs) offer a potentially powerful method to process graph data. However, existing EQGNN models only consider the permutation symmetry of graphs, and failing to fully exploit the geometric and non-geometric information in graphs, resulting in suboptimal performance when processing 3D graph data. To address these limitations, we derive constraints of rotation and permutation equivariance, and then propose a novel rotation- and permutation-equivariant quantum graph neural network (RP-EQGNN). An equivariant module is designed to extract the geometric information. Then, a convolution and entanglement module is constructed to extract non-geometric information. To improve performance of our model, an edge entanglement strategy is designed to perform distinguishable entanglement operations based on edge heterogeneity. The experiment results demonstrate that RP-EQGNN is significantly better for graph regression on the QM9 dataset and the OC20 dataset than Q3DGL and EQC in MAE and achieves results comparable to those of EquiformerV2, Geoformer, SO3KRATES and HEGNN. It also has advantage for point cloud classification on the ModelNet40 dataset over quantum models, including sQCNN-3D and PI-QSVM. RP-EQGNN introduces an innovative approach for processing 3D graph data, establishing a basis for future investigations into symmetries within graph neural networks.
Quantum neural network architectures that have little to no inductive biases are known to face trainability and generalization issues. Inspired by a similar problem, recent breakthroughs in machine learning address this challenge by creating models encoding the symmetries of the learning task. This is materialized through the usage of equivariant neural networks the action of which commutes with that of the symmetry. In this work, we import these ideas to the quantum realm by presenting a comprehensive theoretical framework to design equivariant quantum neural networks (EQNNs) for essentially any relevant symmetry group. We develop multiple methods to construct equivariant layers for EQNNs and analyze their advantages and drawbacks. Our methods can find unitary or general equivariant quantum channels efficiently even when the symmetry group is exponentially large or continuous. As a special implementation, we show how standard quantum convolutional neural networks (QCNNs) can be generalized to group-equivariant QCNNs where both the convolution and pooling layers are equivariant to the symmetry group. We then numerically demonstrate the effectiveness of a S U ( 2 ) -equivariant QCNN over symmetry-agnostic QCNN on a classification task of phases of matter in the bond-alternating Heisenberg model. Our framework can be readily applied to virtually all areas of quantum machine learning. Lastly, we discuss about how symmetry-informed models such as EQNNs provide hopes to alleviate central challenges such as barren plateaus, poor local minima, and sample complexity. Published by the American Physical Society 2024