Abstract Machine learning is among the most widely anticipated use cases for near-term quantum computers, however there remain significant theoretical and implementation challenges impeding its scale up. In particular, there is an emerging body of work which suggests that generic, data agnostic quantum machine learning (QML) architectures may suffer from severe trainability issues, with the gradient of typical variational parameters vanishing exponentially in the number of qubits. Additionally, the high expressibility of QML models can lead to overfitting on training data and poor generalisation performance. A promising strategy to combat both of these difficulties is to construct models which explicitly respect the symmetries inherent in their data, so-called geometric quantum machine learning (GQML). In this work, we utilise the techniques of GQML for the task of image classification, building new QML models which are equivariant with respect to reflections of the images. We find that these networks are capable of consistently and significantly outperforming generic ansatze on complicated real-world image datasets, bringing high-resolution image classification via quantum computers closer to reality. Our work highlights a potential pathway for the future development and implementation of powerful QML models which directly exploit the symmetries of data.
Abstract Machine learning is among the most widely anticipated use cases for near-term quantum computers, however there remain significant theoretical and implementation challenges impeding its scale up. In particular, there is an emerging body of work which suggests that generic, data agnostic quantum machine learning (QML) architectures may suffer from severe trainability issues, with the gradient of typical variational parameters vanishing exponentially in the number of qubits. Additionally, the high expressibility of QML models can lead to overfitting on training data and poor generalisation performance. A promising strategy to combat both of these difficulties is to construct models which explicitly respect the symmetries inherent in their data, so-called geometric quantum machine learning (GQML). In this work, we utilise the techniques of GQML for the task of image classification, building new QML models which are equivariant with respect to reflections of the images. We find that these networks are capable of consistently and significantly outperforming generic ansatze on complicated real-world image datasets, bringing high-resolution image classification via quantum computers closer to reality. Our work highlights a potential pathway for the future development and implementation of powerful QML models which directly exploit the symmetries of data.
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 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 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.
Exploiting the power of quantum computation to realize superior machine learning algorithms has been a major research focus of recent years, but the prospects of quantum machine learning (QML) remain dampened by considerable technical challenges. A particularly significant issue is that generic QML models suffer from so-called barren plateaus in their training landscapes—large regions where cost function gradients vanish exponentially in the number of qubits employed, rendering large models effectively untrainable. A leading strategy for combating this effect is to build problem-specific models that take into account the symmetries of their data in order to focus on a smaller, relevant subset of Hilbert space. In this work, we introduce a family of rotationally equivariant QML models built upon the quantum Fourier transform, and leverage recent insights from the Lie-algebraic study of QML models to prove that (a subset of) our models do not exhibit barren plateaus. In addition to our analytical results we numerically test our rotationally equivariant models on a dataset of simulated scanning tunneling microscope images of phosphorus impurities in silicon, where rotational symmetry naturally arises, and find that they dramatically outperform their generic counterparts in practice. Published by the American Physical Society 2024
To understand the intricacies of quantum geometric machine learning (QGML), it is essential to first examine the foundational concepts of geometric deep learning (GDL). This section addresses key questions such as: “Why is GDL necessary?”, “What advantages does it offer over traditional non-GDL?”, and “How does its complexity compare?”. We also explore the types of data required for GDL models and the preprocessing steps involved, which naturally leads to a discussion on why QGML provides significant advantages over classical GDL. The data commonly used in deep learning is Euclidean (text, image, and audio), meaning it is either one-dimensional or two-dimensional. However, our world is three-dimensional, and so should our data be. Non-Euclidean data represents more complex information with greater accuracy. For instance, in non-Euclidean data, the shortest valid path between two points is not necessarily the Euclidean distance. GDL mainly deals with graphs and manifold data. Graphs consist of nodes and edges – like in a network, where nodes might represent individuals and edges their relationships. Manifold data describes data points that lie on a lower-dimensional surface, even though they are embedded in higher-dimensional space. Our Earth, though not a perfect manifold, is a good approximation for practical purposes, aiding in navigation, remote sensing, and satellite imaging. Given that non-Euclidean data is large scale and irregularly arranged, traditional deep learning methods cannot be applied effectively, necessitating the use of GDL. GDL generally involves graph data processing and manifold data processing. We explore these areas before shifting focus to GQML. In quantum machine learning (QML), inductive biases inductive biases (the choices made during model construction) are crucial. Unlike classical machine learning, which is limited by vectors in Euclidean space, QML operates in Hilbert space. Leveraging underlying symmetries and data structures has led to the rise of QGML. Recent research has focused on designing equivariant quantum neural networks to efficiently find unitary or general equivariant quantum channels, even for large or continuous symmetry groups, alleviating issues like barren plateaus and poor local minima. Efforts have also been made to 2 impose symmetries through classical post-processing and by altering the ansatz to enforce symmetries. Recent studies involve learning symmetries in learning tasks to construct variational quantum learning models with improved generalization performance.