Quantum Spectral Clustering

• URL: http://arxiv.org/abs/2007.00280v4
• Date: Mon, 14 Jun 2021 07:16:18 GMT
• Title: Quantum Spectral Clustering
• Authors: Iordanis Kerenidis, Jonas Landman
• Abstract summary: Spectral clustering is a powerful machine learning algorithm for clustering data with non convex or nested structures. We propose an end-to-end quantum algorithm spectral clustering, extending a number of works in quantum machine learning.
• Score: 5.414308305392762
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Spectral clustering is a powerful unsupervised machine learning algorithm for clustering data with non convex or nested structures. 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 $O(n^3)$, where $n$ is the number of points in the dataset. 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 analogue of the $k$-means algorithm. 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.

Related papers

• Entanglement-Assisted Coding for Arbitrary Linear Computations Over a Quantum MAC [34.32444379837011]
  We study a linear computation problem over a quantum multiple access channel (LC-QMAC) We propose an achievable scheme for LC-QMAC based on the stabilizer formalism and the ideas from entanglement-assisted quantum error-correcting codes (EAQECC)
  arXiv  Detail & Related papers  (2025-01-27T18:35:33Z)
• Quantum Algorithms for Stochastic Differential Equations: A Schrödingerisation Approach [29.662683446339194]
  We propose quantum algorithms for linear differential equations. The gate complexity of our algorithms exhibits an $mathcalO(dlog(Nd))$ dependence on the dimensions. The algorithms are numerically verified for the Ornstein-Uhlenbeck processes, Brownian motions, and one-dimensional L'evy flights.
  arXiv  Detail & Related papers  (2024-12-19T14:04:11Z)
• Quantum multi-row iteration algorithm for linear systems with non-square coefficient matrices [7.174256268278207]
  We propose a quantum algorithm inspired by the classical multi-row iteration method. Our algorithm places less demand on the coefficient matrix, making it suitable for solving inconsistent systems.
  arXiv  Detail & Related papers  (2024-09-06T03:32:02Z)
• Quantum algorithms for Hopcroft's problem [45.45456673484445]
  We study quantum algorithms for Hopcroft's problem which is a fundamental problem in computational geometry. The classical complexity of this problem is well-studied, with the best known algorithm running in $O(n4/3)$ time. Our results are two different quantum algorithms with time complexity $widetilde O(n5/6)$.
  arXiv  Detail & Related papers  (2024-05-02T10:29:06Z)
• Qubit-Efficient Randomized Quantum Algorithms for Linear Algebra [3.4137115855910767]
  We propose a class of randomized quantum algorithms for the task of sampling from matrix functions. Our use of qubits is purely algorithmic, and no additional qubits are required for quantum data structures.
  arXiv  Detail & Related papers  (2023-02-03T17:22:49Z)
• Quantum Clustering with k-Means: a Hybrid Approach [117.4705494502186]
  We design, implement, and evaluate three hybrid quantum k-Means algorithms. We exploit quantum phenomena to speed up the computation of distances. We show that our hybrid quantum k-Means algorithms can be more efficient than the classical version.
  arXiv  Detail & Related papers  (2022-12-13T16:04:16Z)
• Variational Quantum and Quantum-Inspired Clustering [0.0]
  We present a quantum algorithm for clustering data based on a variational quantum circuit. The algorithm allows to classify data into many clusters, and can easily be implemented in few-qubit Noisy Intermediate-Scale Quantum (NISQ) devices.
  arXiv  Detail & Related papers  (2022-06-20T17:02:19Z)
• Quantum K-medians Algorithm Using Parallel Euclidean Distance Estimator [0.0]
  This paper proposes an efficient quantum k-medians clustering algorithm using the powerful quantum Euclidean estimator algorithm. The proposed quantum k-medians algorithm has provided an exponential speed up as compared to the classical version of it.
  arXiv  Detail & Related papers  (2020-12-21T06:38:20Z)
• Quantum algorithms for spectral sums [50.045011844765185]
  We propose new quantum algorithms for estimating spectral sums of positive semi-definite (PSD) matrices. We show how the algorithms and techniques used in this work can be applied to three problems in spectral graph theory.
  arXiv  Detail & Related papers  (2020-11-12T16:29:45Z)
• Laplacian Eigenmaps with variational circuits: a quantum embedding of graph data [0.0]
  We propose a method to compute a Laplacian Eigenmap using a quantum variational circuit. Tests on 32 nodes graph with a quantum simulator show that we can achieve similar performances as the classical laplacian eigenmap algorithm.
  arXiv  Detail & Related papers  (2020-11-10T14:51:25Z)
• Quantum Gram-Schmidt Processes and Their Application to Efficient State Read-out for Quantum Algorithms [87.04438831673063]
  We present an efficient read-out protocol that yields the classical vector form of the generated state. Our protocol suits the case that the output state lies in the row space of the input matrix. One of our technical tools is an efficient quantum algorithm for performing the Gram-Schmidt orthonormal procedure.
  arXiv  Detail & Related papers  (2020-04-14T11:05:26Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.