A quantum algorithm for solving eigenproblem of the Laplacian matrix of a fully connected weighted graph

• URL: http://arxiv.org/abs/2203.14451v2
• Date: Sat, 28 May 2022 14:58:15 GMT
• Title: A quantum algorithm for solving eigenproblem of the Laplacian matrix of a fully connected weighted graph
• Authors: Hai-Ling Liu, Su-Juan Qin, Lin-Chun Wan, Chao-Hua Yu, Shi-Jie Pan, Fei Gao, and Qiao-Yan Wen
• Abstract summary: We propose an efficient quantum algorithm to solve the eigenproblem of the Laplacian matrix of a fully connected weighted graph. Specifically, we adopt the optimal Hamiltonian simulation technique based on the block-encoding framework. We also show that our algorithm can be extended to solve the eigenproblem of symmetric (non-symmetric) normalized Laplacian matrix.
• Score: 4.045204834863644
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Solving eigenproblem of the Laplacian matrix of a fully connected weighted graph has wide applications in data science, machine learning, and image processing, etc. However, this is very challenging because it involves expensive matrix operations. Here, we propose an efficient quantum algorithm to solve it based on a assumption that the element of each vertex and its norms can be effectively accessed via a quantum random access memory data structure. Specifically, we adopt the optimal Hamiltonian simulation technique based on the block-encoding framework to implement the quantum simulation of the Laplacian matrix. Then, the eigenvalues and eigenvectors of the Laplacian matrix are extracted by the quantum phase estimation algorithm. The core of our entire algorithm is to construct the block-encoding of the Laplacian matrix. To achieve this, we propose in detail how to construct the block-encodings of operators containing the information of the weight matrix and the degree matrix respectively, and further obtain the block-encoding of the Laplacian matrix. Compared with its classical counterpart, our algorithm has a polynomial speedup on the number of vertices and an exponential speedup on the dimension of each vertex. We also show that our algorithm can be extended to solve the eigenproblem of symmetric (non-symmetric) normalized Laplacian matrix.

Related papers

• A square-root speedup for finding the smallest eigenvalue [0.6597195879147555]
  We describe a quantum algorithm for finding the smallest eigenvalue of a Hermitian matrix. This algorithm combines Quantum Phase Estimation and Quantum Amplitude Estimation to achieve a quadratic speedup. We also provide a similar algorithm with the same runtime that allows us to prepare a quantum state lying mostly in the matrix's low-energy subspace.
  arXiv  Detail & Related papers  (2023-11-07T22:52:56Z)
• Vectorization of the density matrix and quantum simulation of the von Neumann equation of time-dependent Hamiltonians [65.268245109828]
  We develop a general framework to linearize the von-Neumann equation rendering it in a suitable form for quantum simulations. We show that one of these linearizations of the von-Neumann equation corresponds to the standard case in which the state vector becomes the column stacked elements of the density matrix. A quantum algorithm to simulate the dynamics of the density matrix is proposed.
  arXiv  Detail & Related papers  (2023-06-14T23:08:51Z)
• A hybrid quantum-classical algorithm for multichannel quantum scattering of atoms and molecules [62.997667081978825]
  We propose a hybrid quantum-classical algorithm for solving the Schr"odinger equation for atomic and molecular collisions. The algorithm is based on the $S$-matrix version of the Kohn variational principle, which computes the fundamental scattering $S$-matrix. We show how the algorithm could be scaled up to simulate collisions of large polyatomic molecules.
  arXiv  Detail & Related papers  (2023-04-12T18:10:47Z)
• 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 algorithms for matrix operations and linear systems of equations [65.62256987706128]
  We propose quantum algorithms for matrix operations using the "Sender-Receiver" model. These quantum protocols can be used as subroutines in other quantum schemes.
  arXiv  Detail & Related papers  (2022-02-10T08:12:20Z)
• Quantum Algorithms based on the Block-Encoding Framework for Matrix Functions by Contour Integrals [1.5293427903448018]
  We show a framework to implement the linear combination of the inverses on quantum computers. We propose a quantum algorithm for matrix functions based on the framework.
  arXiv  Detail & Related papers  (2021-06-15T12:10:35Z)
• Quantum algorithms for powering stable Hermitian matrices [0.7734726150561088]
  Matrix powering is a fundamental computational primitive in linear algebra. We present two quantum algorithms that can achieve speedup over the classical matrix powering algorithms.
  arXiv  Detail & Related papers  (2021-03-15T12:20:04Z)
• 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)
• Multi-View Spectral Clustering with High-Order Optimal Neighborhood Laplacian Matrix [57.11971786407279]
  Multi-view spectral clustering can effectively reveal the intrinsic cluster structure among data. This paper proposes a multi-view spectral clustering algorithm that learns a high-order optimal neighborhood Laplacian matrix. Our proposed algorithm generates the optimal Laplacian matrix by searching the neighborhood of the linear combination of both the first-order and high-order base.
  arXiv  Detail & Related papers  (2020-08-31T12:28:40Z)
• Optimal Iterative Sketching with the Subsampled Randomized Hadamard Transform [64.90148466525754]
  We study the performance of iterative sketching for least-squares problems. We show that the convergence rate for Haar and randomized Hadamard matrices are identical, andally improve upon random projections. These techniques may be applied to other algorithms that employ randomized dimension reduction.
  arXiv  Detail & Related papers  (2020-02-03T16:17:50Z)

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.