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.
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.
Recent advances in classical machine learning have shown that creating models with inductive biases encoding the symmetries of a problem can greatly improve performance. Importation of these ideas, combined with an existing rich body of work at the nexus of quantum theory and symmetry, has given rise to the field of Geometric Quantum Machine Learning (GQML). Following the success of its classical counterpart, it is reasonable to expect that GQML will play a crucial role in developing problem-specific and quantum-aware models capable of achieving a computational advantage. Despite the simplicity of the main idea of GQML -- create architectures respecting the symmetries of the data -- its practical implementation requires a significant amount of knowledge of group representation theory. We present an introduction to representation theory tools from the optics of quantum learning, driven by key examples involving discrete and continuous groups. These examples are sewn together by an exposition outlining the formal capture of GQML symmetries via "label invariance under the action of a group representation", a brief (but rigorous) tour through finite and compact Lie group representation theory, a reexamination of ubiquitous tools like Haar integration and twirling, and an overview of some successful strategies for detecting symmetries.
Geometric Machine Learning (GML) successes have been achieved through the thorough study and design of new equivariant neural networks. In comparison, geometric quantum machine learning (GQML) models lack such a detailed understanding and, despite already several proposals, a unifying perspective on their design remains elusive. In this work, we focus on GQML models for graph problems that showcase a lot of structure and still remain frontier in machine learning. For the case when n-node graphs are encoded in n-qubit states, we provide a comprehensive characterization of their constituents. Taken together, these furnish us with a toolbox for the design of quantum graph models, and we further probe its benefits including the natural integration with classical models, generalization of known GQML models (sometimes extending their expressivity at virtually no cost), and straightforward classical pre-training strategies. The latter two features are demonstrated in dedicated numerical experiments.
Geometric Machine Learning (GML) has shown that respecting non-Euclidean geometry in data spaces can significantly improve performance over naive Euclidean assumptions. In parallel, Quantum Machine Learning (QML) has emerged as a promising paradigm that leverages superposition, entanglement, and interference within quantum state manifolds for learning tasks. This paper offers a unifying perspective by casting QML as a specialized yet more expressive branch of GML. We argue that quantum states, whether pure or mixed, reside on curved manifolds (e.g., projective Hilbert spaces or density-operator manifolds), mirroring how covariance matrices inhabit the manifold of symmetric positive definite (SPD) matrices or how image sets occupy Grassmann manifolds. However, QML also benefits from purely quantum properties, such as entanglement-induced curvature, that can yield richer kernel structures and more nuanced data embeddings. We illustrate these ideas with published and newly discussed results, including hybrid classical -quantum pipelines for diabetic foot ulcer classification and structural health monitoring. Despite near-term hardware limitations that constrain purely quantum solutions, hybrid architectures already demonstrate tangible benefits by combining classical manifold-based feature extraction with quantum embeddings. We present a detailed mathematical treatment of the geometrical underpinnings of quantum states, emphasizing parallels to classical Riemannian geometry and manifold-based optimization. Finally, we outline open research challenges and future directions, including Quantum Large Language Models (LLMs), quantum reinforcement learning, and emerging hardware approaches, demonstrating how synergizing GML and QML principles can unlock the next generation of machine intelligence.
Quantum Machine Learning (QML) and Quantum Deep Learning (QDL) are emerging research areas that combine quantum computing with machine learning and deep learning techniques.These approaches aim to improve computational speed, feature representation, optimisation, and learning performance in complex data-driven problems.This review paper presents a structured discussion of recent developments in QML and QDL, focusing on major algorithms, application areas, implementation approaches, current limitations, and future research directions.The review covers important quantum-based learning methods, including quantum support vector machines, quantum k-nearest neighbours, quantum principal component analysis, quantum neural networks, variational quantum classifiers, quantum convolutional neural networks, quantum reinforcement learning, quantum generative models, and hybrid quantum-classical models.The paper also discusses how these methods are being explored in healthcare, finance, cybersecurity, communication networks, image processing, natural language processing, and optimisation.However, the review shows that most QML and QDL models are still in the early stages of research.Current challenges include noisy quantum hardware, limited qubit count, data encoding issues, circuit design problems, difficulties with model training, and a lack of standardised benchmarks.The review concludes that QML and QDL have strong future potential, but their practical success depends on improved hardware, stronger algorithms, reliable frameworks, and fair comparison with classical models.
Quantum machine learning sits at the intersection of two paradigm-shifting technologies: quantum computing and deep learning. By exploiting quantum mechanical phenomena – superposition, entanglement, and interference – QML promises exponential or quadratic speedups over classical algorithms for certain learning tasks, as well as the ability to model quantum data natively. However, the path to practical quantum advantage remains contested, constrained by noisy intermediate-scale quantum (NISQ) hardware, limited qubit counts (50–1,000), coherence times (50–500 μs), and high gate error rates (10⁻³–10⁻⁴). This paper provides a rigorous, technically deep survey of QML: mathematical foundations (Hilbert spaces, unitary evolution, measurement, quantum circuits), canonical algorithms (quantum kernel methods, variational quantum circuits (VQCs), quantum neural networks (QNNs), QSVMs, quantum generative adversarial networks (QGANs), encoding strategies (angle, amplitude, basis, and Hamiltonian encoding), optimization landscapes (barren plateaus, parameter shift rules, ansatz design), hardware aware implementation on superconducting (IBM, Google), trapped ion (IonQ), and neutral atom (Pasqal) platforms, empirical benchmarks on near term devices for classification, generative modeling, and quantum chemistry, and pathways to fault tolerant quantum machine learning (FTQML) with error correction (surface codes, logical qubits). Researchers critically assess claims of quantum advantage, identify scenarios where QML may outperform classical ML (small data, high-dimensional feature spaces from quantum systems, kernel estimation), and propose concrete evaluation metrics (quantum-classical cross-validation, resource estimation).
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.