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) 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.
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.
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 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.
In hierarchal order of molecular geometry, we compare the performances of Geometric Quantum Machine Learning models. Two molecular datasets are considered: the simplistic linear shaped LiH-molecule and the trigonal pyramidal molecule NH3. Both accuracy and generalizability metrics are considered. A classical equivariant model is used as a baseline for the performance comparison. The comparative performance of Quantum Machine Learning models with no symmetry equivariance, rotational and permutational equivariance, and graph embedded permutational equivariance is investigated. The performance differentials and the molecular geometry in question reveals the criteria for choice of models for generalizability. Graph embedding of features is shown to be an effective pathway to greater trainability for geometric datasets. Permutational symmetric embedding is found to be the most generalizable quantum Machine Learning model for geometric learning.
Quantum computing offers a promising avenue to reduce growing machine learning model complexity as required in large language models and simulation models for weather forecasts, financial forecasts, or engineering.Graph neural networks are a particular class of machine learning models that have garnered much attention for their ability to deal well with structured data.We investigate how to enhance existing GNNs and find through the inductive bias that quantum circuits are used best to encode node features.The proposed Quantum Feature Embeddings (QFEs) turn raw input features into quantum states, enabling non-linear and entangled representations.In particular, QFEs provide normalized, non-redundant weight matrices in an exponentially larger feature space and require much fewer qubits than fully quantum graph neural networks.On standard graph benchmark datasets, we showcase that for the same parameter count QFEs perform better than their classical counterpart, and are able to match the performance of an exponentially larger model.Finally, we study the potential benefit of using a hybrid quantum graph neural network over a classic alternative on a concrete use case, laser cutting.We find that the proposed model has the performance and thus the near-term potential to uplift these business applications.