A Probabilistic framework for Quantum Clustering

2001

We propose a novel clustering method that is based on physical intuition derived from quantum mechanics. Starting with given data points, we construct a scale-space probability function. Viewing the latter as the lowest eigenstate of a Schrodinger equation, we use simple analytic operations to derive a potential function whose minima determine cluster centers. The method has one parameter, determining the

Mathematics, 2017

Data clustering is a vital tool for data analysis. This work shows that some existing useful methods in data clustering are actually based on quantum mechanics and can be assembled into a~powerful and accurate data clustering method where the efficiency of computational quantum chemistry eigenvalue methods is therefore applicable. These methods can be applied to scientific data, engineering data and even text.

Advances in Neural Information Processing Systems 14, 2002

We propose a novel clustering method that is an extension of ideas inherent to scale-space clustering and support-vector clustering. Like the latter, it associates every data point with a vector in Hilbert space, and like the former it puts emphasis on their total sum, that is equal to the scalespace probability function. The novelty of our approach is the study of an operator in Hilbert space, represented by the Schrödinger equation of which the probability function is a solution. This Schrödinger equation contains a potential function that can be derived analytically from the probability function. We associate minima of the potential with cluster centers. The method has one variable parameter, the scale of its Gaussian kernel. We demonstrate its applicability on known data sets. By limiting the evaluation of the Schrödinger potential to the locations of data points, we can apply this method to problems in high dimensions.

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.

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.

Physical Review Letters, 2001

We propose a novel clustering method that is based on physical intuition derived from quantum mechanics. Starting with given data points, we construct a scale-space probability function. Viewing the latter as the lowest eigenstate of a Schrödinger equation, we use simple analytic operations to derive a potential function whose minima determine cluster centers. The method has one parameter, determining the scale over which cluster structures are searched. We demonstrate it on data analyzed in two dimensions (chosen from the eigenvectors of the correlation matrix). The method is applicable in higher dimensions by limiting the evaluation of the Schrödinger potential to the locations of data points.

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.

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&...

International Journal of Data Mining, Modelling and Management, 2010

The emerging field of quantum computing has recently created much interest in the computer science community due to the new concepts it suggests to store and process data. In this paper, we explore some of these concepts to cope with the data clustering problem. Data clustering is a key task for most fields like data mining and pattern recognition. It aims to discover cohesive groups in large datasets. In our work, we cast this problem as an optimisation process and we describe a novel framework, which relies on a quantum representation to encode the search space and a quantum evolutionary search strategy to optimise a quality measure in quest of a good partitioning of the dataset. Results on both synthetic and real data are very promising and show the ability of the method to identify valid clusters and also its effectiveness comparing to other evolutionary algorithms. . Her areas of interest include computational intelligence, quantum inspired computing, bioinformatics and image analysis.

Physical Review X

We introduce the problem of unsupervised classification of quantum data, namely, of systems whose quantum states are unknown. We derive the optimal single-shot protocol for the binary case, where the states in a disordered input array are of two types. Our protocol is universal and able to automatically sort the input under minimal assumptions, yet partially preserving information contained in the states. We quantify analytically its performance for arbitrary size and dimension of the data. We contrast it with the performance of its classical counterpart, which clusters data that has been sampled from two unknown probability distributions. We find that the quantum protocol fully exploits the dimensionality of the quantum data to achieve a much higher performance, provided data is at least three-dimensional. For the sake of comparison, we discuss the optimal protocol when the classical and quantum states are known.