Quantum Clustering

arXiv (Cornell University), 2023

In this paper, two novel measurement-based clustering algorithms are proposed based on quantum parallelism and entanglement. The Euclidean distance metric is used as a measure of 'similarity' between the data points. The first algorithm follows a divisive approach and the bound for each cluster is determined based on the number of ancillae used to label the clusters. The second algorithm is based on unsharp measurements where we construct the set of effect operators with a gaussian probability distribution to cluster similar data points. We specifically implemented the algorithm on a concentric circle data set for which the classical clustering approach fails. It is found that the presented clustering algorithms perform better than the classical divisive one; both in terms of clustering and time complexity which is found to be O(kN logN) for the first and O(N 2) for the second one. Along with that we also implemented the algorithm on the Churrtiz data set of cities and the Wisconsin breast cancer dataset where we found an accuracy of approximately 97.43% which For the later case is achieved by the appropriate choice of the variance of the gaussian window.

IEEE International Conference on Image Processing 2005, 2005

This paper introduces a new nonparametric estimation approach that can be used for data that is not necessarily Gaussian distributed. The proposed approach employs the Shrödinger partial differential equation. We assume that each data sample is associated with a quantum physics particle that has a radial field around its value. We consider a statistical estimation approach for finding the size of the influence field around each data sample. By implementing the Shrödinger equation we obtain a potential field that is assimilated with the data density. The regions of minima in the potential are determined by calculating the local Hessian on the potential hypersurface. The quantum clustering approach is applied for blind separation of signals and for segmenting SAR images of terrain based on surface normal orientation.

Physical Review E, 2009

A given set of data-points in some feature space may be associated with a Schrödinger equation whose potential is determined by the data. This is known to lead to good clustering solutions. Here we extend this approach into a full-fledged dynamical scheme using a time-dependent Schrödinger equation. Moreover, we approximate this Hamiltonian formalism by a truncated calculation within a set of Gaussian wave functions (coherent states) centered around the original points. This allows for analytic evaluation of the time evolution of all such states, opening up the possibility of exploration of relationships among data-points through observation of varying dynamical-distances among points and convergence of points into clusters. This formalism may be further supplemented by preprocessing, such as dimensional reduction through singular value decomposition or feature filtering. PACS numbers: 89.75.Fb,89.90.+n,95.75.Pq,89.75.Kd −1/2 i,j

arXiv: Quantum Physics, 2018

We present an algorithm for quantum-assisted cluster analysis (QACA) that makes use of the topological properties of a D-Wave 2000Q quantum processing unit (QPU). Clustering is a form of unsupervised machine learning, where instances are organized into groups whose members share similarities. The assignments are, in contrast to classification, not known a priori, but generated by the algorithm. We explain how the problem can be expressed as a quadratic unconstrained binary optimization (QUBO) problem, and show that the introduced quantum-assisted clustering algorithm is, regarding accuracy, equivalent to commonly used classical clustering algorithms. Quantum annealing algorithms belong to the class of metaheuristic tools, applicable for solving binary optimization problems. Hardware implementations of quantum annealing, such as the quantum annealing machines produced by D-Wave Systems, have been subject to multiple analyses in research, with the aim of characterizing the technology&...

arXiv (Cornell University), 2022

Quantum computing is a promising paradigm based on quantum theory for performing 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 degree 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 can be more efficient than the classical version, still obtaining comparable clustering results.

Quantum clustering (QC), is a data clustering algorithm based on quantum mechanics which is accomplished by substituting each point in a given dataset with a Gaussian. The width of the Gaussian is a value, a hyper-parameter which can be manually defined and manipulated to suit the application. Numerical methods are used to find all the minima of the quantum potential as they correspond to cluster centers. Herein, we investigate the mathematical task of expressing and finding all the roots of the exponential polynomial corresponding to the minima of a two-dimensional quantum potential. This is an outstanding task because normally such expressions are impossible to solve analytically. However, we prove that if the points are all included in a square region of size , there is only one minimum. This bound is not only useful in the number of solutions to look for, by numerical means, it allows to to propose a new numerical approach "per block". This technique decreases the number of particles (or samples) by approximating some groups of particles to weighted particles. These findings are not only useful to the quantum clustering problem but also for the exponential polynomials encountered in quantum chemistry, Solid-state Physics and other applications.

Physica A: Statistical Mechanics and its Applications, 2001

We discuss novel clustering methods that are based on mapping data points to a Hilbert space by means of a Gaussian kernel. The ÿrst method, support vector clustering (SVC), searches for the smallest sphere enclosing data images in Hilbert space. The second, quantum clustering (QC), searches for the minima of a potential function deÿned in such a Hilbert space. In SVC, the minimal sphere, when mapped back to data space, separates into several components, each enclosing a separate cluster of points. A soft margin constant helps in coping with outliers and overlapping clusters. In QC, minima of the potential deÿne cluster centers, and equipotential surfaces are used to construct the clusters. In both methods, the width of the Gaussian kernel controls the scale at which the data are probed for cluster formations. We demonstrate the performance of the algorithms on several data sets.

Mathematics

In real-world scenarios, identifying the optimal number of clusters in a dataset is a difficult task due to insufficient knowledge. Therefore, the indispensability of sophisticated automatic clustering algorithms for this purpose has been contemplated by some researchers. Several automatic clustering algorithms assisted by quantum-inspired metaheuristics have been developed in recent years. However, the literature lacks definitive documentation of the state-of-the-art quantum-inspired metaheuristic algorithms for automatically clustering datasets. This article presents a brief overview of the automatic clustering process to establish the importance of making the clustering process automatic. The fundamental concepts of the quantum computing paradigm are also presented to highlight the utility of quantum-inspired algorithms. This article thoroughly analyses some algorithms employed to address the automatic clustering of various datasets. The reviewed algorithms were classified accord...

arXiv: Medical Physics, 2021

Data clustering has been widely used in data analysis and classification. In the present work, a method based on dynamic quantum clustering is proposed for the segmentation and analysis of brain tumor MRI. The results open the possibility of applications to multi modality medical imaging.

2008