Abstract Quantum state clustering provides a powerful tool for quantum information processing. This paper focuses on the classification of single-qubit pure and mixed states via clustering algorithms, and discovers that the classical K -means algorithm has two limitations in clustering quantum states: the Euclidean distance cannot accurately reflect the similarity between quantum states, and the centroid update based on the arithmetic mean frequently fails to yield valid quantum states. To address these issues, we propose two distance metrics for quantum states: the dynamic distance and the Bloch distance, and adopt the K -medoids method to update the centroid, ensuring that the cluster center always remains a valid quantum state, thereby designing a quantum-data-adaptive clustering algorithm. Experimental results show that the dynamic distance achieves a clustering accuracy as high as 91.4 % , but at the cost of higher computational complexity. In contrast, the Bloch distance significantly reduces computational complexity while maintaining clustering performance comparable to that of the dynamic distance, achieving a clustering accuracy of 87.4 % . This paper provides a practical method for achieving efficient and reliable clustering of single-qubit quantum states.
Abstract Quantum state clustering provides a powerful tool for quantum information processing. This paper focuses on the classification of single-qubit pure and mixed states via clustering algorithms, and discovers that the classical K -means algorithm has two limitations in clustering quantum states: the Euclidean distance cannot accurately reflect the similarity between quantum states, and the centroid update based on the arithmetic mean frequently fails to yield valid quantum states. To address these issues, we propose two distance metrics for quantum states: the dynamic distance and the Bloch distance, and adopt the K -medoids method to update the centroid, ensuring that the cluster center always remains a valid quantum state, thereby designing a quantum-data-adaptive clustering algorithm. Experimental results show that the dynamic distance achieves a clustering accuracy as high as 91.4 % , but at the cost of higher computational complexity. In contrast, the Bloch distance significantly reduces computational complexity while maintaining clustering performance comparable to that of the dynamic distance, achieving a clustering accuracy of 87.4 % . This paper provides a practical method for achieving efficient and reliable clustering of single-qubit quantum states.
Quantum computing, based on quantum theory, holds great promise as an advanced computational paradigm for achieving fast computations. Quantum algorithms are expected to surpass their classical counterparts in terms of computational complexity for certain tasks, including machine learning. In this paper, we design, implement, and evaluate three hybrid quantum k-Means algorithms, exploiting different degrees of parallelism. Indeed, each algorithm incrementally leverages quantum parallelism to reduce the complexity of the cluster assignment step up to a constant cost. In particular, we exploit quantum phenomena to speed up the computation of distances. The core idea is that the computation of distances between records and centroids can be executed simultaneously, thus saving time, especially for big datasets. We show that our hybrid quantum k-Means algorithms are theoretically faster than the classical algorithm, while experiments suggest that it is possible to obtain comparable clustering results.
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.
Recently, the volume of the generated data has increased significantly, leading to the need for computational techniques capable of handling such data efficiently. As a result, many quantum algorithms have been developed, and the domain of quantum machine learning (QML) has become more extensive. Traditional k-means algorithms for quantum computers usually make use of deep or shallow quantum circuits, resulting in sub-optimal clustering results. In our work, we introduce two types of quantum-inspired k-means algorithm: (1) the quantum subtraction operation and (2) the rotational-difference dissimilarity measure. Our second framework measures the dissimilarity using a quantum rotation-based dissimilarity circuit, which encodes the relative difference of the states. We employ a Hadamard-test-based circuit design, as well as an alternative technique that is not used in other quantum dissimilarities like the swap test. The introduced algorithms are evaluated on six different datasets—Iris, Wine, Breast Cancer Wisconsin, Blobs, Moons, and the noisier Iris dataset. Evaluation involves clustering validity metrics, as well as the classic classification performance metrics. The findings show that the rotational-based dissimilarity metric allows us to obtain clustering results comparable with the results obtained by the classical counterpart, thus showing the feasibility of the introduced distance calculation technique.
It is getting harder to deal with the large data sets by the classical hierarchical clustering algorithm, so we propose an efficient quantum hierarchical clustering algorithm, in which the quantum bit (qubit) is used to represent the data point in the space.For quantum entanglement, the distance between two data points is calculated through adding an auxiliary particle to construct the entangled state.Then a projective measurement is performed on the auxiliary particle alone.The distance between two points is acquired by the projective measurement.We use the distance of the cluster centroids as a measure of similarity between clusters.Also, based on the principle of the minimum cluster centroids distance, the nearest two clusters are merged.We aim at improving time and space complexity and effect of the clustering of the hierarchical clustering algorithm.
The classical K-means clustering algorithm is widely used in various fields due to its simple implementation and efficient computation, but the classical K-means clustering algorithm relies on the random selection of the initial center of mass, which is prone to fall into the deadlock of local optimality. In order to break through this limitation, the quantum K-means clustering algorithm is introduced, which is able to explore multiple potential clustering center combinations at the same time through the parallelism of quantum computation, so as to have a greater probability of converging to the globally optimal solution. Quantum K-means clustering algorithms typically employ fidelity as a similarity measure between quantum states, and similarity is assessed by calculating the probability of overlap between quantum states. However, the fidelity only quantizes the pure state information of the quantum states and ignores the classical statistical features of the data itself, which may lead to unreasonable clustering boundaries in mixed state or noise interference scenarios. In response to the above problems, this paper proposes an improved quantum-classical hybrid similarity metric, whose core idea is to incorporate the dual constraints of quantum information and classical features.
This study introduces a Hybrid Quantum K-Means Clustering Algorithm with automatic cluster detection for classifying cancerous and non-cancerous gene expression data. The method employs Quantum Multi-Feature Mapping for state encoding, Swap Test-based quantum distance estimation, and Quantum Gradient-Based Optimization to dynamically identify the optimal number of clusters by minimizing intra-cluster variance. Initial centroids are selected through a probability-proportional distance strategy, improving stability and accuracy. Applied to breast cancer datasets, the approach surpasses the existing quantum K-Means algorithm, achieving a Silhouette Score of 0.641 (compared to 0.601), a Calinski-Harabasz Index of 766.57 (compared to 617.65), and a Davies-Bouldin Index of 0.659 (compared to 0.704). These results indicate superior cluster compactness and separation. Although the proposed algorithm exhibits slightly higher time complexity O (N×Kmax×Mobs) due to iterative optimization, it significantly outperforms predefined-K quantum K-Means in clustering accuracy, error reduction, and practical feasibility. Its efficiency in handling high-dimensional data and resilience to quantum noise highlights its potential for real-world bioinformatics applications, particularly in cancer classification using gene expression profiles.