Resolvent-based quantum phase estimation: Towards estimation of parametrized eigenvalues

• URL: http://arxiv.org/abs/2410.04837v2
• Date: Mon, 01 Sep 2025 17:13:25 GMT
• Title: Resolvent-based quantum phase estimation: Towards estimation of parametrized eigenvalues
• Authors: Abhijeet Alase, Salini Karuvade,
• Abstract summary: We propose a novel approach for estimating the eigenvalues of non-normal matrices.<n>We construct the first efficient algorithm for estimating the phases of the unimodular eigenvalues of a given non-unitary matrix.<n>We then construct an efficient algorithm for estimating the real eigenvalues of a given non-Hermitian matrix.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Quantum algorithms for estimating the eigenvalues of matrices, including the phase estimation algorithm, serve as core subroutines in a wide range of quantum algorithms, including those in quantum chemistry and quantum machine learning. The standard quantum eigenvalue (phase) estimation algorithm accepts a Hermitian (unitary) matrix and a state in an unknown superposition of its eigenstates as input, and coherently records the estimates for real eigenvalues (eigenphases) in an ancillary register. Extension of quantum eigenvalue and phase estimation algorithms to the case of non-normal input matrices is obstructed by several factors such as non-orthogonality of eigenvectors, existence of generalized eigenvectors and the fact that eigenvalues may lie anywhere on the complex plane. In this work, we propose a novel approach for estimating the eigenvalues of non-normal matrices based on preparation of a state that we call the "resolvent state". We construct the first efficient algorithm for estimating the phases of the unimodular eigenvalues of a given non-unitary matrix. We then construct an efficient algorithm for estimating the real eigenvalues of a given non-Hermitian matrix, achieving complexities that match the best known results while operating under significantly relaxed assumptions on the non-real part of the spectrum. The resolvent-based approach that we introduce also extends to estimating eigenvalues that lie on a parametrized complex curve, subject to explicitly stated conditions, thereby paving the way for a new paradigm of parametric eigenvalue estimation.

Related papers

• Quantum-Inspired Algorithm for Classical Spin Hamiltonians Based on Matrix Product Operators [0.18665975431697432]
  We propose a tensor-network (TN) approach for solving classical optimization problems inspired by spectral filtering and sampling on quantum states.<n>We represent the transformed Hamiltonian as a matrix product operator (MPO) and form an immense power of this object via truncated MPO-MPO contractions.<n>In contrast to the density-matrix renormalization group, our approach provides a straightforward route to systematic improvement by increasing the bond dimension and is better at avoiding local minima.
  arXiv  Detail & Related papers  (2026-02-05T02:29:37Z)
• Heisenberg-Limited Quantum Eigenvalue Estimation for Non-normal Matrices [5.733109475878588]
  Estimating the eigenvalues of non-normal matrices is a foundational problem with far-reaching implications.<n>Here we introduce a new class of quantum algorithms that directly address this challenge.<n>Our work lays the foundation for a rigorous and scalable quantum computing approach to one of the most demanding problems in linear algebra.
  arXiv  Detail & Related papers  (2025-10-22T14:55:44Z)
• Near-optimal Rank Adaptive Inference of High Dimensional Matrices [46.66027208538566]
  We address the problem of estimating a high-dimensional matrix from linear measurements.<n>We propose an algorithm that combines a Least-Squares estimator with a universal singular value thresholding procedure.<n>Our results rely on an enhanced analysis of matrix denoising methods based on singular value thresholding.
  arXiv  Detail & Related papers  (2025-10-09T12:01:46Z)
• Quantum algorithm for solving generalized eigenvalue problems with application to the Schrödinger equation [0.0]
  Estimating excited-state energies is challenging for classical algorithms due to exponential scaling with system size.<n>We present a quantum algorithm for estimating eigenvalues and singular values of parameterized matrix families.
  arXiv  Detail & Related papers  (2025-06-16T14:24:30Z)
• Spectral Estimation with Free Decompression [47.81955761814048]
  We introduce a novel method of "free decompression" to estimate the spectrum of very large (impalpable) matrices.<n>Our method can be used to extrapolate from the empirical spectral densities of small submatrices to infer the eigenspectrum of extremely large (impalpable) matrices.
  arXiv  Detail & Related papers  (2025-06-13T17:49:25Z)
• Quantum Hermitian conjugate and encoding unnormalized matrices [49.494595696663524]
  We develop the family of matrix-manipulation algorithms based on the encoding the matrix elements into the probability amplitudes of the pure superposition state of a certain quantum system. We introduce two extensions to these algorithms which allow (i) to perform Hermitian conjugation of matrices under consideration and (ii) to weaken the restriction to the absolute values of matrix elements unavoidably imposed by the normalization condition for a pure quantum state.
  arXiv  Detail & Related papers  (2025-03-27T08:49:59Z)
• Quantum Eigensolver for Non-Normal Matrices via Ground State Energy Estimation [0.4511923587827302]
  Large-scale eigenvalue problems pose a significant challenge to classical computers.<n>We propose a quantum algorithm that outputs an estimate of an eigenvalue to within additive error $epsilon$ with probability at least $1-p_rm fail$.<n>Our algorithm is the first general eigenvalue algorithm that achieves this scaling.
  arXiv  Detail & Related papers  (2025-02-25T11:43:47Z)
• Simulating NMR Spectra with a Quantum Computer [49.1574468325115]
  This paper provides a formalization of the complete procedure of the simulation of a spin system's NMR spectrum. We also explain how to diagonalize the Hamiltonian matrix with a quantum computer, thus enhancing the overall process's performance.
  arXiv  Detail & Related papers  (2024-10-28T08:43:40Z)
• Entrywise error bounds for low-rank approximations of kernel matrices [55.524284152242096]
  We derive entrywise error bounds for low-rank approximations of kernel matrices obtained using the truncated eigen-decomposition. A key technical innovation is a delocalisation result for the eigenvectors of the kernel matrix corresponding to small eigenvalues. We validate our theory with an empirical study of a collection of synthetic and real-world datasets.
  arXiv  Detail & Related papers  (2024-05-23T12:26:25Z)
• Improving Expressive Power of Spectral Graph Neural Networks with Eigenvalue Correction [55.57072563835959]
  spectral graph neural networks are characterized by filters. We propose an eigenvalue correction strategy that can free filters from the constraints of repeated eigenvalue inputs.
  arXiv  Detail & Related papers  (2024-01-28T08:12:00Z)
• 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)
• $h(1) \oplus su(2)$ vector algebra eigenstates with eigenvalues in the matrix domain [0.0]
  We find a subset of generalized vector coherent states in the matrix domain. For a special choice of the matrix eigenvalue parameters we found the so-called vector coherent states with matrices associated to the Heisenberg-Weyl group.
  arXiv  Detail & Related papers  (2023-01-25T18:10:01Z)
• On confidence intervals for precision matrices and the eigendecomposition of covariance matrices [20.20416580970697]
  This paper tackles the challenge of computing confidence bounds on the individual entries of eigenvectors of a covariance matrix of fixed dimension. We derive a method to bound the entries of the inverse covariance matrix, the so-called precision matrix. As an application of these results, we demonstrate a new statistical test, which allows us to test for non-zero values of the precision matrix.
  arXiv  Detail & Related papers  (2022-08-25T10:12:53Z)
• Contour Integral-based Quantum Algorithm for Estimating Matrix Eigenvalue Density [5.962184741057505]
  We propose a quantum algorithm for computing the eigenvalue density in a given interval. The eigenvalue count in a given interval is derived as the probability of observing a bit pattern in a fraction of the qubits of the output state.
  arXiv  Detail & Related papers  (2021-12-10T08:58:44Z)
• Minimax Estimation of Linear Functions of Eigenvectors in the Face of Small Eigen-Gaps [95.62172085878132]
  Eigenvector perturbation analysis plays a vital role in various statistical data science applications. We develop a suite of statistical theory that characterizes the perturbation of arbitrary linear functions of an unknown eigenvector. In order to mitigate a non-negligible bias issue inherent to the natural "plug-in" estimator, we develop de-biased estimators.
  arXiv  Detail & Related papers  (2021-04-07T17:55:10Z)
• Structured eigenvalue problems in electronic structure methods from a unified perspective [0.0]
  The quaternion matrix eigenvalue problem and the linear response (Bethe-Salpeter) eigenvalue problem for excitation energies are frequently encountered structured eigenvalue problems. We show that the identification of Lie group structures for their eigenvectors provides a framework to design diagonalization algorithms.
  arXiv  Detail & Related papers  (2020-09-02T15:20:15Z)
• Understanding Implicit Regularization in Over-Parameterized Single Index Model [55.41685740015095]
  We design regularization-free algorithms for the high-dimensional single index model. We provide theoretical guarantees for the induced implicit regularization phenomenon.
  arXiv  Detail & Related papers  (2020-07-16T13:27:47Z)
• Eigendecomposition-Free Training of Deep Networks for Linear Least-Square Problems [107.3868459697569]
  We introduce an eigendecomposition-free approach to training a deep network. We show that our approach is much more robust than explicit differentiation of the eigendecomposition. Our method has better convergence properties and yields state-of-the-art results.
  arXiv  Detail & Related papers  (2020-04-15T04:29:34Z)
• Relative Error Bound Analysis for Nuclear Norm Regularized Matrix Completion [101.83262280224729]
  We develop a relative error bound for nuclear norm regularized matrix completion. We derive a relative upper bound for recovering the best low-rank approximation of the unknown matrix.
  arXiv  Detail & Related papers  (2015-04-26T13:12:16Z)

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.