Approximate Graph Spectral Decomposition with the Variational Quantum Eigensolver

• URL: http://arxiv.org/abs/1912.12366v1
• Date: Fri, 27 Dec 2019 23:27:38 GMT
• Title: Approximate Graph Spectral Decomposition with the Variational Quantum Eigensolver
• Authors: Josh Payne and Mario Srouji
• Abstract summary: Spectral graph theory studies the relationships between the eigenvectors and eigenvalues of Laplacian and adjacency matrices and their associated graphs. The Variational Quantum Eigensolver (VQE) algorithm was proposed as a hybrid quantum/classical algorithm. In this paper, we will expand upon the VQE algorithm to analyze the spectra of directed and undirected graphs.
• Score: 1.0152838128195465
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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 Eigensolver (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 $M\in \mathbb{R}^{n\times n}$. 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.

Related papers

• Quantum walk informed variational algorithm design [0.0]
  We present a theoretical framework for the analysis of amplitude transfer in Quantum Variational Algorithms (QVAs) We develop algorithms for unconstrained and constrained novel optimisations. We show significantly improved convergence over preexisting QVAs.
  arXiv  Detail & Related papers  (2024-06-17T15:04:26Z)
• Gaussian Entanglement Measure: Applications to Multipartite Entanglement of Graph States and Bosonic Field Theory [50.24983453990065]
  An entanglement measure based on the Fubini-Study metric has been recently introduced by Cocchiarella and co-workers. We present the Gaussian Entanglement Measure (GEM), a generalization of geometric entanglement measure for multimode Gaussian states. By providing a computable multipartite entanglement measure for systems with a large number of degrees of freedom, we show that our definition can be used to obtain insights into a free bosonic field theory.
  arXiv  Detail & Related papers  (2024-01-31T15:50:50Z)
• Quantum eigenvalue processing [0.0]
  Problems in linear algebra can be solved on a quantum computer by processing eigenvalues of the non-normal input matrices. We present a Quantum EigenValue Transformation (QEVT) framework for applying arbitrary transformations on eigenvalues of block-encoded non-normal operators. We also present a Quantum EigenValue Estimation (QEVE) algorithm for operators with real spectra.
  arXiv  Detail & Related papers  (2024-01-11T19:49:31Z)
• Using Variational Eigensolvers on Low-End Hardware to Find the Ground State Energy of Simple Molecules [0.0]
  Key properties of physical systems can be described by the eigenvalues of matrices that represent the system. Computational algorithms that determine the eigenvalues of these matrices exist, but they generally suffer from a loss of performance as the matrix grows in size. This process can be expanded to quantum computation to find the eigenvalues with better performance than the classical algorithms.
  arXiv  Detail & Related papers  (2023-10-29T18:36:18Z)
• 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)
• Graph Polynomial Convolution Models for Node Classification of Non-Homophilous Graphs [52.52570805621925]
  We investigate efficient learning from higher-order graph convolution and learning directly from adjacency matrix for node classification. We show that the resulting model lead to new graphs and residual scaling parameter. We demonstrate that the proposed methods obtain improved accuracy for node-classification of non-homophilous parameters.
  arXiv  Detail & Related papers  (2022-09-12T04:46:55Z)
• Stochastic Parallelizable Eigengap Dilation for Large Graph Clustering [12.544602297450533]
  We aim to identify clusters of nodes where most edges fall within clusters and only few edges fall between clusters. A core step of spectral clustering is performing an eigendecomposition of the corresponding graph Laplacian matrix. This paper introduces a parallelizable approach to dilating the spectrum in order to accelerate SVD solvers and in turn, spectral clustering.
  arXiv  Detail & Related papers  (2022-07-29T10:13:07Z)
• A quantum algorithm for solving eigenproblem of the Laplacian matrix of a fully connected weighted graph [4.045204834863644]
  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.
  arXiv  Detail & Related papers  (2022-03-28T02:24:08Z)
• Quantum Algorithms for Data Representation and Analysis [68.754953879193]
  We provide quantum procedures that speed-up the solution of eigenproblems for data representation in machine learning. The power and practical use of these subroutines is shown through new quantum algorithms, sublinear in the input matrix's size, for principal component analysis, correspondence analysis, and latent semantic analysis. Results show that the run-time parameters that do not depend on the input's size are reasonable and that the error on the computed model is small, allowing for competitive classification performances.
  arXiv  Detail & Related papers  (2021-04-19T00:41:43Z)
• Joint Network Topology Inference via Structured Fusion Regularization [70.30364652829164]
  Joint network topology inference represents a canonical problem of learning multiple graph Laplacian matrices from heterogeneous graph signals. We propose a general graph estimator based on a novel structured fusion regularization. We show that the proposed graph estimator enjoys both high computational efficiency and rigorous theoretical guarantee.
  arXiv  Detail & Related papers  (2021-03-05T04:42:32Z)
• 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)

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.