Quantum Eigensolver with Exponentially Improved Dependence on Parameters

• URL: http://arxiv.org/abs/2502.18119v1
• Date: Tue, 25 Feb 2025 11:43:47 GMT
• Title: Quantum Eigensolver with Exponentially Improved Dependence on Parameters
• Authors: Honghong Lin, Yun Shang,
• Abstract summary: Eigenvalue estimation is a fundamental problem in numerical analysis and scientific computation.<n>This work proposes an efficient quantum eigenvalues based on quantum transformations of the matrix.
• Score: 0.8057006406834466
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Eigenvalue estimation is a fundamental problem in numerical analysis and scientific computation. The case of complex eigenvalues is considered to be hard. This work proposes an efficient quantum eigensolver based on quantum polynomial transformations of the matrix. Specifically, we construct Chebyshev polynomials and positive integer power functions transformations to the input matrix within the framework of block encoding. Our algorithm simply employs Hadamard test on the matrix polynomials to generate classical data, and then pocesses the data to retrieve the information of the eigenvalue. The algorithmic complexity depends logarithmically on precision and failure probability, and is independent of the matrix size. Therefore, the algorithm provides exponential advantage over previous work.

Related papers

• Matrix encoding method in variational quantum singular value decomposition [49.494595696663524]
  Conditional measurement is involved to avoid small success probability in ancilla measurement. The objective function for the algorithm can be obtained probabilistically via measurement of the state of a one-qubit subsystem.
  arXiv  Detail & Related papers  (2025-03-19T07:01:38Z)
• Laplace transform based quantum eigenvalue transformation via linear combination of Hamiltonian simulation [13.96848357202551]
  We propose an efficient quantum algorithm for performing a class of eigenvalue transformations that can be expressed as a certain type of matrix Laplace transformation. We demonstrate that our eigenvalue transformation approach can solve this problem without explicitly inverting $A$.
  arXiv  Detail & Related papers  (2024-11-06T15:47:48Z)
• Efficient conversion from fermionic Gaussian states to matrix product states [48.225436651971805]
  We propose a highly efficient algorithm that converts fermionic Gaussian states to matrix product states. It can be formulated for finite-size systems without translation invariance, but becomes particularly appealing when applied to infinite systems. The potential of our method is demonstrated by numerical calculations in two chiral spin liquids.
  arXiv  Detail & Related papers  (2024-08-02T10:15:26Z)
• Purely quantum algorithms for calculating determinant and inverse of matrix and solving linear algebraic systems [43.53835128052666]
  We propose quantum algorithms for the computation of the determinant and inverse of an $(N-1) times (N-1)$ matrix.<n>This approach is a straightforward modification of the existing algorithm for determining the determinant of an $N times N$ matrix.<n>We provide suitable circuit designs for all three algorithms, each estimated to require $O(N log N)$ in terms of space.
  arXiv  Detail & Related papers  (2024-01-29T23:23:27Z)
• 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)
• 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)
• Automatic and effective discovery of quantum kernels [41.61572387137452]
  Quantum computing can empower machine learning models by enabling kernel machines to leverage quantum kernels for representing similarity measures between data.<n>We present an approach to this problem, which employs optimization techniques, similar to those used in neural architecture search and AutoML.<n>The results obtained by testing our approach on a high-energy physics problem demonstrate that, in the best-case scenario, we can either match or improve testing accuracy with respect to the manual design approach.
  arXiv  Detail & Related papers  (2022-09-22T16:42:14Z)
• 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 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)
• A Quantum Computer Amenable Sparse Matrix Equation Solver [0.0]
  We study problems involving the solution of matrix equations, for which there currently exists no efficient, general quantum procedure. We develop a generalization of the Harrow/Hassidim/Lloyd algorithm by providing an alternative unitary for eigenphase estimation. This unitary has the advantage of being well defined for any arbitrary matrix equation, thereby allowing the solution procedure to be directly implemented on quantum hardware.
  arXiv  Detail & Related papers  (2021-12-05T15:42:32Z)
• Robust 1-bit Compressive Sensing with Partial Gaussian Circulant Matrices and Generative Priors [54.936314353063494]
  We provide recovery guarantees for a correlation-based optimization algorithm for robust 1-bit compressive sensing. We make use of a practical iterative algorithm, and perform numerical experiments on image datasets to corroborate our results.
  arXiv  Detail & Related papers  (2021-08-08T05:28:06Z)
• 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.