We consider the problem of distinguishing two vectors (visualized as images or barcodes) and learning if they are related to one another. For this, we develop a geometric quantum machine learning (GQML) approach with embedded symmetries that allows for the classification of similar and dissimilar pairs based on global correlations, and enables generalization from just a few samples. Unlike GQML algorithms developed to date, we propose to focus on symmetry-aware measurement adaptation that outperforms unitary parametrizations. We compare GQML for similarity testing against classical deep neural networks and convolutional neural networks with Siamese architectures. We show that quantum networks demonstrate strong empirical performance gains over their classical counterparts in generalization. We explain this difference in performance by analyzing correlated distributions used for composing our dataset. We relate the similarity testing to problems that showcase a proven maximal separation between the bounded-error quantum polynomial time complexity class and the polynomial hierarchy. While the ability to achieve a formal advantage depends on how data are loaded, we discuss how similar problems can benefit from quantum machine learning. Finally, we present a 40-qubit hardware implementation of our quantum model, using a superconducting processor (IBM Kyiv), showing remarkable scalability and resilience to noise.
We consider the problem of distinguishing two vectors (visualized as images or barcodes) and learning if they are related to one another. For this, we develop a geometric quantum machine learning (GQML) approach with embedded symmetries that allows for the classification of similar and dissimilar pairs based on global correlations, and enables generalization from just a few samples. Unlike GQML algorithms developed to date, we propose to focus on symmetry-aware measurement adaptation that outperforms unitary parametrizations. We compare GQML for similarity testing against classical deep neural networks and convolutional neural networks with Siamese architectures. We show that quantum networks demonstrate strong empirical performance gains over their classical counterparts in generalization. We explain this difference in performance by analyzing correlated distributions used for composing our dataset. We relate the similarity testing to problems that showcase a proven maximal separation between the bounded-error quantum polynomial time complexity class and the polynomial hierarchy. While the ability to achieve a formal advantage depends on how data are loaded, we discuss how similar problems can benefit from quantum machine learning. Finally, we present a 40-qubit hardware implementation of our quantum model, using a superconducting processor (IBM Kyiv), showing remarkable scalability and resilience to noise.
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.
Geometric quantum machine learning of BQPA protocols and latent graph classifiers
Geometric quantum machine learning (GQML) aims to embed problem symmetries for learning efficient solving protocols. However, the question remains if (G)QML can be routinely used for constructing protocols with an exponential separation from classical analogs. In this Letter we consider Simon's problem for learning properties of Boolean functions, and show that this can be related to an unsupervised circuit classification problem. Using the workflow of geometric QML, we learn from first principles Simon's algorithm, thus discovering an example of BQPA=BPP protocol with respect to some dataset (oracle A). Our key findings include the development of an equivariant feature map for embedding Boolean functions, based on twirling with respect to identified bitflip and permutational symmetries, and measurement based on invariant observables with a sampling advantage. The proposed workflow points to the importance of data embeddings and classical post-processing, while keeping the variational circuit as a trivial identity operator. Next, developing the intuition for the function learning, we visualize instances as directed computational hypergraphs, and observe that the GQML protocol can access their global topological features for distinguishing bijective and surjective functions. Finally, we discuss the prospects for learning other BQPA-type protocols, and conjecture that this depends on the ability of simplifying embeddings-based oracles A applied as a linear combination of unitaries.
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.
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.