Abstract Clustering high-dimensional data with inherent geometric symmetries is a key challenge in unsupervised learning. While traditional algorithms like K-Means struggle with complex data structures, existing quantum methods often lack equivariance and face scalability issues on Noisy Intermediate-Scale Quantum (NISQ) hardware. This creates a need for a clustering approach that can effectively handle structured data. We introduce the p4m Equivariant Quantum Clustering (EQC) framework, a novel approach that integrates equivariant quantum neural networks with quantum kernel methods. Our model utilizes an 8-qubit quantum circuit with p4m symmetry-preserving operations to ensure robust feature extraction. We employ a hybrid quantum-classical optimization strategy to tune the model parameters. We evaluated the EQC framework on the MNIST and Quark-Gluon datasets using a noise-free quantum simulator. Under simulated depolarizing noise, EQC maintains stable performance on MNIST (Silhouette Score: 0.471; Accuracy: 53.3%), demonstrating robustness suitable for near-term quantum hardware. On the Quark-Gluon dataset, EQC achieved a Silhouette Score of 0.512 and 74.8% accuracy, a significant improvement over classical K-Means (Silhouette Score: 0.183; Accuracy: 58.4%). The model also demonstrated state-of-the-art performance on MNIST, achieving a Silhouette Score of 0.557 and 76.3% accuracy, consistently surpassing other quantum and classical baselines. Although current evaluations are based on simulated backends, the inclusion of realistic noise modeling highlights EQC’s resilience and scalability potential. These results suggest that incorporating symmetry-aware design principles in quantum machine learning can offer tangible advantages for structured data analysis on NISQ-era systems.
Abstract Clustering high-dimensional data with inherent geometric symmetries is a key challenge in unsupervised learning. While traditional algorithms like K-Means struggle with complex data structures, existing quantum methods often lack equivariance and face scalability issues on Noisy Intermediate-Scale Quantum (NISQ) hardware. This creates a need for a clustering approach that can effectively handle structured data. We introduce the p4m Equivariant Quantum Clustering (EQC) framework, a novel approach that integrates equivariant quantum neural networks with quantum kernel methods. Our model utilizes an 8-qubit quantum circuit with p4m symmetry-preserving operations to ensure robust feature extraction. We employ a hybrid quantum-classical optimization strategy to tune the model parameters. We evaluated the EQC framework on the MNIST and Quark-Gluon datasets using a noise-free quantum simulator. Under simulated depolarizing noise, EQC maintains stable performance on MNIST (Silhouette Score: 0.471; Accuracy: 53.3%), demonstrating robustness suitable for near-term quantum hardware. On the Quark-Gluon dataset, EQC achieved a Silhouette Score of 0.512 and 74.8% accuracy, a significant improvement over classical K-Means (Silhouette Score: 0.183; Accuracy: 58.4%). The model also demonstrated state-of-the-art performance on MNIST, achieving a Silhouette Score of 0.557 and 76.3% accuracy, consistently surpassing other quantum and classical baselines. Although current evaluations are based on simulated backends, the inclusion of realistic noise modeling highlights EQC’s resilience and scalability potential. These results suggest that incorporating symmetry-aware design principles in quantum machine learning can offer tangible advantages for structured data analysis on NISQ-era systems.
Privacy-preserving clustering is critical for analyzing sensitive data in healthcare, cybersecurity, and enterprise applications, where maintaining data confidentiality must be balanced with analytical performance. This paper presents Equivariant Quantum Clustering (EQC), a parameter-efficient framework that integrates symmetry-aware quantum circuits with differential privacy to improve the privacy-utility tradeoff. EQC employs p4m equivariant parameter sharing to reduce circuit complexity while preserving informative feature representations. The framework is evaluated on three privacy-sensitive datasets: NSL-KDD, CERT Insider Threat v6.2, and a synthetic MIMIC-III clinical dataset. On the NSL-KDD benchmark, EQC achieves 79.3% clustering accuracy while reducing membership inference attack success to 38.3% under a privacy budget of ε = 1.0 and δ = 10^-5, outperforming representative classical and quantum baselines. Ablation studies indicate that the performance gains primarily arise from parameter-efficient circuit design combined with differential privacy. The results demonstrate that EQC provides a practical quantum-ready framework for secure and privacy-preserving clustering across heterogeneous sensitive datasets.
Clustering remains a cornerstone of unsupervised learning, yet classical algorithms often struggle with the inherent complexities of modern datasets, such as high dimensionality, non-linear separability, and significant noise. This paper introduces the Hybrid Quantum-Classical Clustering Algorithm (HQCA), a novel framework that leverages the power of quantum computation and the robustness of classical machine learning to overcome these limitations. HQCA integrates a quantum kernel estimation phase, utilizing a parameterized quantum circuit to project data into a high-dimensional Hilbert space, thereby enhancing cluster separability where classical kernels fail. Furthermore, it employs a quantum-enhanced optimization routine, based on the Quantum Approximate Optimization Algorithm (QAOA), to determine optimal cluster centroids, thereby mitigating the common pitfall of local minima encountered by algorithms like K-Means. The algorithm also incorporates a quantum-inspired gap statistic to autonomously determine the optimal number of clusters. We rigorously evaluate HQCA against state-of-the-art classical clustering methods, including K-Means, DBSCAN, and Spectral Clustering, on multiple real-world benchmark datasets from the UCI repository, such as Iris, Wine, and a high-dimensional image dataset (Fashion-MNIST). Our results demonstrate that HQCA consistently achieves superior performance across multiple evaluation metrics, including Purity, Silhouette Score, and Adjusted Rand Index (ARI), particularly excelling in scenarios involving complex, non-linear data manifolds. The proposed hybrid paradigm not only signifies a substantial leap in clustering accuracy and robustness but also provides a scalable pathway for leveraging current Noisy Intermediate-Scale Quantum (NISQ) devices in practical data analytics workflows, marking a critical step toward potential quantum advantage in unsupervised learning. We emphasize that the demonstrated advantage is algorithmic and representational, not an asymptotic runtime advantage.
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.
Clustering is one of the most fundamental tasks in machine learning, and the k-means clustering algorithm is perhaps one of the most widely used clustering algorithms. However, it suffers from several limitations, such as sensitivity to centroid initialization, difficulty capturing non-linear structure, and poor performance in high-dimensional spaces. Recent work has proposed improved initialization strategies and quantum-assisted distance computation, but the similarity metric itself has largely remained classical. In this study, we propose a quantum-enhanced variant of k-means that replaces the Euclidean distance with a quantum kernel derived from the inner product between feature-mapped quantum states. Using the Iris dataset, we use multiple quantum feature maps, including entangled SU2 and ZZ circuits, to embed classical data into a higher-dimensional Hilbert space where cluster structures become more separable. We will also be testing using another dataset, namely the breast cancer dataset. Similarity between data points is computed through the inner product between two states. Our results show that this approach achieves improved clustering stability and competitive accuracy compared to the classical algorithm, with the SU2 feature map yielding an accuracy of 88.6 % on the Iris dataset and 91.0 % on the breast cancer dataset, despite operating on NISQ-feasible shallow circuits. These findings suggest that quantum kernels provide a richer similarity landscape than traditional distance metrics, offering a promising path toward more robust unsupervised learning in the NISQ era.
The rapid data surge from the high-luminosity Large Hadron Collider introduces critical computational challenges requiring novel approaches for efficient data processing in particle physics. Quantum machine learning, with its capability to leverage the extensive Hilbert space of quantum hardware, offers a promising solution. However, current quantum graph neural networks (GNNs) lack robustness to noise and are often constrained by fixed symmetry groups, limiting adaptability in complex particle interaction modeling. This article demonstrates that replacing the classical Lorentz group equivariant block modules in LorentzNet with a dressed quantum circuit significantly enhances performance despite using≈5.5times fewer parameters. Additionally, quantum circuits effectively replace MLPs by inherently preserving symmetries, with Lorentz symmetry integration ensuring robust handling of relativistic invariance. OurLorentz-equivariantquantumgraphneuralnetwork (Lorentz-EQGNN) achieved 74.00% test accuracy and an AUC of 87.38% on the Quark-Gluon jet tagging dataset, outperforming the classical and quantum GNNs with a reduced architecture using only 4 qubits. On the electron–photon dataset, Lorentz-EQGNN reached 67.00% test accuracy and an AUC of 68.20%, demonstrating competitive results with just 800 training samples. Evaluation of our model on generic MNIST and FashionMNIST datasets confirmed Lorentz-EQGNN’s efficiency, achieving 88.10% and 74.80% test accuracy, respectively. Ablation studies validated the impact of quantum components on performance, with notable improvements in background rejection rates over classical counterparts. These results highlight Lorentz-EQGNN’s potential for immediate applications in noise-resilient jet tagging, event classification, and broader data-scarce HEP tasks.
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.