Strange Bedfellows: Quantum Mechanics and Data Mining

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), 2013

How does one search for a needle in a multi-dimensional haystack without knowing what a needle is and without knowing if there is one in the haystack? This kind of problem requires a paradigm shift-away from hypothesis driven searches of the data-towards a methodology that lets the data speak for itself. Dynamic Quantum Clustering (DQC) is such a methodology. DQC is a powerful visual method that works with big, high-dimensional data. It exploits variations of the density of the data (in feature space) and unearths subsets of the data that exhibit correlations among all the measured variables. The outcome of a DQC analysis is a movie that shows how and why sets of data-points are eventually classified as members of simple clusters or as members of-what we call-extended structures. This allows DQC to be successfully used in a non-conventional exploratory mode where one searches data for unexpected information without the need to model the data. We show how this works for big, complex, real-world datasets that come from five distinct fields: i.e., x-ray nano-chemistry, condensed matter, biology, seismology and finance. These studies show how DQC excels at uncovering unexpected, small-but meaningful-subsets of the data that contain important information. We also establish an important new result: namely, that big, complex datasets often contain interesting structures that will be missed by many conventional clustering techniques. Experience shows that these structures appear frequently enough that it is crucial to know they can exist, and that when they do, they encode important hidden information. In short, we not only demonstrate that DQC can be flexibly applied to datasets that present significantly different challenges, we also show how a simple analysis can be used to look for the needle in the haystack, determine what it is, and find what this means.

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), 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.

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.

International Journal of Theoretical Physics, 2017

We address the problem of binary classification by using a quantum version of the Nearest Mean Classifier (NMC). Our proposal is indeed an advanced version of previous one (see Sergioli et al. 2017 that i) is able to be naturally generalized to arbitrary number of features and ii) exhibits better performances with respect to the classical NMC for several datasets. Further, we show that the quantum version of NMC is not invariant under rescaling. This allows us to introduce a free parameter, i.e. the rescaling factor, that could be useful to get a further improvement of the classification performance.

International Journal of Data Analysis Techniques and Strategies, 2013

In previous work, a novel approach to data clustering based on quantum evolutionary algorithm has been proposed. In a comparison to other evolutionary clustering algorithms, the approach showed a high performance in terms of effectiveness and quality of found clusters. Although the approach is sound, it tends to be trapped in local minima, which slows the convergence. The approach is based on degrees of belonging having a fixed relationship with the distance between the data points and the clusters. The fixed relationship ignores completely the dataset distribution. In this paper, we modify the approach to improve its convergence. We also modify the function calculating the degrees of belonging by taking inspiration from possibilistic clustering. Comparison has been done with approaches based on degrees of belonging like fuzzy, possibilistic, hybrid fuzzy possibilistic clustering and other quantum evolutionary algorithm. Results on both real and synthetic datasets show that the modifications brought to the approach enable a more efficient exploration of the search space which improves the convergence speed and quality.

2008

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.

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

Machine Learning, 2013

We show how the quantum paradigm can be used to speed up unsupervised learning algorithms. More precisely, we explain how it is possible to accelerate learning algorithms by quantizing some of their subroutines. Quantization refers to the process that partially or totally converts a classical algorithm to its quantum counterpart in order to improve performance. In particular, we give quantized versions of clustering via minimum spanning tree, divisive clustering and k-medians that are faster than their classical analogues. We also describe a distributed version of k-medians that allows the participants to save on the global communication cost of the protocol compared to the classical version. Finally, we design quantum algorithms for the construction of a neighbourhood graph, outlier detection as well as smart initialization of the cluster centres. Keywords Unsupervised learning • Clustering • Quantum learning • Quantum information processing • Grover's algorithm 1 Introduction Consider the following scenario, which illustrates a highly challenging clustering task. Imagine that you are an employee of the Department of Statistics of the United Nations. Your boss gives you the demographic data of all the Earth inhabitants and asks you to anal

2020 International Joint Conference on Neural Networks (IJCNN), 2020

Recently, more researchers are interested in the domain of quantum machine learning as it can manipulate and classify large numbers of vectors in high dimensional space in reasonable time. In this paper, we propose a new approach called Quantum Collaborative K-means which is based on combining several clustering models based on quantum K-means. This collaboration consists of exchanging the information of each algorithm locally in order to find a common underlying structure for clustering. Comparing the classical version of collaborative clustering to our approach, we notice that we have an exponential speed up: while the classical version takes O(K × L × M × N), the quantum version takes only O(K×L×log(M ×N)). And comparing to the quantum version of K-means, we get a better solution in terms of the criteria of validation which means in terms of clustering. The empirical evaluations validate the benefits of the proposed approach.

This paper proposes a new quantum-like method for the binary classification applied to classical datasets. Inspired by the quantum Helstrom measurement, this innovative approach has enabled us to define a new classifier, called Helstrom Quantum Centroid (HQC). This binary classifier (inspired by the concept of distinguishability between quantum states) acts on density matrices-called density patterns-that are the quantum encoding of classical patterns of a dataset. In this paper we compare the performance of HQC with respect to twelve standard (linear and non-linear) classifiers over fourteen different datasets. The experimental results show that HQC outperforms the other classifiers when compared to the Balanced Accuracy and other statistical measures. Finally, we show that the performance of our classifier is positively correlated to the increase in the number of "quantum copies" of a pattern and the resulting tensor product thereof.

arXiv (Cornell University), 2020

Counting the number of clusters, when these clusters overlap significantly is a challenging problem in machine learning. We argue that a purely mathematical quantum theory, formulated using the path integral technique, when applied to non-physics modeling leads to non-physics quantum theories that are statistical in nature. We show that a quantum theory can be a more robust statistical theory to separate data to count overlapping clusters. The theory is also confirmed from data simulations. This works identify how quantum theory can be effective in counting clusters and hope to inspire the field to further apply such techniques.

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.