Spectral graph partitioning is a well known technique to estimate clusters in undirected graphs. Recent approaches explored efficient spectral algorithms for directed and mixed graphs utilizing various matrix representations. Despite its success in clustering tasks, classical spectral algorithms suffer from a cubic growth in runtime. In this paper, we propose a quantum spectral clustering algorithm for discovering clusters and properties of mixed graphs. Our experimental results based on numerical simulations demonstrate that our quantum spectral clustering outperforms classical spectral clustering techniques. Specifically, our approach leads to a linear growth in complexity, while state-of-the-art classical counterpart leads to cubic growth. In a case study, we apply our proposed algorithm to preform unsupervised machine learning using both real and simulated quantum computers. This work opens an avenue for efficient implementation of machine learning algorithms on directed as well as mixed graphs by making use of the inherent potential quantum speedup.
Spectral graph partitioning is a well known technique to estimate clusters in undirected graphs. Recent approaches explored efficient spectral algorithms for directed and mixed graphs utilizing various matrix representations. Despite its success in clustering tasks, classical spectral algorithms suffer from a cubic growth in runtime. In this paper, we propose a quantum spectral clustering algorithm for discovering clusters and properties of mixed graphs. Our experimental results based on numerical simulations demonstrate that our quantum spectral clustering outperforms classical spectral clustering techniques. Specifically, our approach leads to a linear growth in complexity, while state-of-the-art classical counterpart leads to cubic growth. In a case study, we apply our proposed algorithm to preform unsupervised machine learning using both real and simulated quantum computers. This work opens an avenue for efficient implementation of machine learning algorithms on directed as well as mixed graphs by making use of the inherent potential quantum speedup.
Spectral clustering is a powerful unsupervised machine learning algorithm for clustering data with nonconvex or nested structures [A. Y. Ng, M. I. Jordan, and Y. Weiss, On spectral clustering: Analysis and an algorithm, in Advances in Neural Information Processing Systems 14: Proceedings of the 2001 Conference (MIT Press, Cambridge, MA, 2002), pp. 849--856]. With roots in graph theory, it uses the spectral properties of the Laplacian matrix to project the data in a low-dimensional space where clustering is more efficient. Despite its success in clustering tasks, spectral clustering suffers in practice from a fast-growing running time of O(n3), where n is the number of points in the data set. In this work we propose an end-to-end quantum algorithm performing spectral clustering, extending a number of works in quantum machine learning. The quantum algorithm is composed of two parts: the first is the efficient creation of the quantum state corresponding to the projected Laplacian matrix, and the second consists of applying the existing quantum analog of the k-means algorithm [I. Kerenidis, J. Landman, A. Luongo, and A. Prakash, q-means: A quantum algorithm for unsupervised machine learning, in Advances in Neural Information Processing Systems 32: Proceedings of the 2019 Conference (Curran Associates, Red Hook, NY, 2020), pp. 4136--4146]. Both steps depend polynomially on the number of clusters, as well as precision and data parameters arising from quantum procedures, and polylogarithmically on the dimension of the input vectors. Our numerical simulations show an asymptotic linear growth with n when all terms are taken into account, significantly better than the classical cubic growth. This work opens the path to other graph-based quantum machine learning algorithms, as it provides routines for efficient computation and quantum access to the incidence, adjacency, and projected Laplacian matrices of a graph.
Spectral clustering is one of the most robust methods of an unsupervised machine learning field. Compared with the well-known k-means, spectral clustering uses the spectral properties of the Laplacian matrix to project data to a low-dimensional clustering with higher efficiency. However, for a sparse network with a substantial degree of heterogeneity, standard spectral clustering often overfits noise in the periphery of a sparse and stochastic graph. In our work, we combined the quantum version of the k-means algorithm with regularized spectral clustering and proposed a new quantum machine learning algorithm. It is more like a combination of classical and quantum algorithms. Compared with the established quantum k-means algorithm, our method has obvious computational advantages in non-convex or nested structures. At this stage, the research of quantum algorithms mainly stays in the simulation of classical computers. Our method does not need to expend a lot of effort to preprocess the data at the initial quantum state stage. Under the assumption of quantum data encoding, we theoretically demonstrate the feasibility of the quantum regularized spectral clustering algorithm. Then combine Q-means algorithm complexity advantages to complete the clustering task. This work aims to pave the path and help realize quantum spectral clustering algorithms on real quantum computers in the future and help create other versions of quantum machine learning algorithms. We provide simulations and data examples to illustrate these results.
Spectral graph theory is a branch of mathematics that studies the relationships between the eigenvectors and eigenvalues of Laplacian and adjacency matrices and their associated graphs. The Variational Quantum Eigen- solver (VQE) algorithm was proposed as a hybrid quantum/classical algorithm that is used to quickly determine the ground state of a Hamiltonian, and more generally, the lowest eigenvalue of a matrix M ∈ Rnxn. There are many interesting problems associated with the spectral decompositions of associated matrices, such as partitioning, embedding, and the determination of other properties. In this paper, we will expand upon the VQE algorithm to analyze the spectra of directed and undirected graphs. We evaluate runtime and accuracy comparisons (empirically and theoretically) between different choices of ansatz parameters, graph sizes, graph densities, and matrix types, and demonstrate the effectiveness of our approach on Rigetti's QCS platform on graphs of up to 64 vertices, finding eigenvalues of adjacency and Laplacian matrices. We finally make direct comparisons to classical performance with the Quantum Virtual Machine (QVM) in the appendix, observing a superpolynomial runtime improvement of our algorithm when run using a quantum computer.*
Given an undirected, unweighted graph with n vertices and m edges, the maximum cut problem is to find a partition of the n vertices into disjoint subsets V1 and V2 such that the number of edges between them is as large as possible. Classically, it is an NP-complete problem, which has potential applications ranging from circuit layout design, statistical physics, computer vision, machine learning and network science to clustering. In this paper, we propose a quantum algorithm to solve the maximum cut problem for any graph G with a quadratic speedup over its classical counterparts, where the temporal and spatial complexities are reduced to, respectively, O(2n/r) and O(m2). With respect to oracle-related quantum algorithms for NP-complete problems, we identify our algorithm as optimal. Furthermore, to justify the feasibility of the proposed algorithm, we successfully solve a typical maximum cut problem for a graph with three vertices and two edges by carrying out experiments on IBM's quantum computer.
The exponential growth of data and the increasing complexity of high-dimensional classification problems have pushed classical computing methods to their limits. Quantum computing emerges as a promising paradigm to address these challenges. This research article explores advanced quantum algorithms for big data clustering and high-dimensional classification. We investigate quantum versions of K-means, spectral clustering, and support vector machines, comparing their performance with classical counterparts. Our results demonstrate significant speedups in processing time and improvements in clustering quality for high-dimensional datasets. Additionally, we propose a novel quantum-inspired classical algorithm that bridges the gap between quantum and classical approaches. This comprehensive study provides insights into the potential of quantum computing in revolutionizing data analysis and machine learning, paving the way for future advancements in the field.
In this thesis, we investigate whether quantum algorithms can be used in the field of machine learning for both long and near term quantum computers. We will first recall the fundamentals of machine learning and quantum computing and then describe more precisely how to link them through linear algebra: we introduce quantum algorithms to efficiently solve tasks such as matrix product or distance estimation. These results are then used to develop new quantum algorithms for unsupervised machine learning, such as k-means and spectral clustering. This allows us to define many fundamental procedures, in particular in vector and graph analysis. We will also present new quantum algorithms for neural networks, or deep learning. For this, we introduce an algorithm to perform a quantum convolution product on images, as well as a new way to perform a fast tomography on quantum states. We prove that these quantum algorithms are faster versions of equivalent classical algorithms, but exhibit random effects due to the quantum nature of the computation. Many simulations have been carried out to study these effects and measure their learning accuracy on real data. Finally, we will present a quantum orthogonal neural network circuit adapted to the currently available small and imperfect quantum computers. This allows us to perform real experiments to test our theory.