Quantum algorithms for the generalized eigenvalue problem
- URL: http://arxiv.org/abs/2112.02554v3
- Date: Sun, 6 Mar 2022 10:49:14 GMT
- Title: Quantum algorithms for the generalized eigenvalue problem
- Authors: Jin-Min Liang, Shu-Qian Shen, Ming Li, Shao-Ming Fei
- Abstract summary: generalized eigenvalue (GE) problems are of particular importance in various areas of science engineering and machine learning.
We present a variational quantum algorithm for finding the desired generalized eigenvalue of the GE problem, $mathcalA|psirangle=lambdamathcalB|psirangle$, by choosing suitable loss functions.
We numerically implement our algorithms to conduct a 2-qubit simulation and successfully find the generalized eigenvalues of the matrix pencil $(mathcalA,,mathcalB)$
- Score: 6.111964049119244
- License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
- Abstract: The generalized eigenvalue (GE) problems are of particular importance in
various areas of science engineering and machine learning. We present a
variational quantum algorithm for finding the desired generalized eigenvalue of
the GE problem, $\mathcal{A}|\psi\rangle=\lambda\mathcal{B}|\psi\rangle$, by
choosing suitable loss functions. Our approach imposes the superposition of the
trial state and the obtained eigenvectors with respect to the weighting matrix
$\mathcal{B}$ on the Rayleigh-quotient. Furthermore, both the values and
derivatives of the loss functions can be calculated on near-term quantum
devices with shallow quantum circuit. Finally, we propose a full quantum
generalized eigensolver (FQGE) to calculate the minimal generalized eigenvalue
with quantum gradient descent algorithm. As a demonstration of the principle,
we numerically implement our algorithms to conduct a 2-qubit simulation and
successfully find the generalized eigenvalues of the matrix pencil
$(\mathcal{A},\,\mathcal{B})$. The numerically experimental result indicates
that FQGE is robust under Gaussian noise.
Related papers
- Improved quantum algorithm for calculating eigenvalues of differential operators and its application to estimating the decay rate of the perturbation distribution tail in stochastic inflation [0.0]
We propose a quantum algorithm for estimating the first eigenvalue of a differential operator $mathcalL$ on $mathbbRd$.
We then consider the application of our method to a problem in a theoretical framework for cosmic inflation known as quantum inflation.
arXiv Detail & Related papers (2024-10-03T07:56:20Z) - Calculating response functions of coupled oscillators using quantum phase estimation [40.31060267062305]
We study the problem of estimating frequency response functions of systems of coupled, classical harmonic oscillators using a quantum computer.
Our proposed quantum algorithm operates in the standard $s-sparse, oracle-based query access model.
We show that a simple adaptation of our algorithm solves the random glued-trees problem in time.
arXiv Detail & Related papers (2024-05-14T15:28:37Z) - 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) - Quantum tomography of helicity states for general scattering processes [55.2480439325792]
Quantum tomography has become an indispensable tool in order to compute the density matrix $rho$ of quantum systems in Physics.
We present the theoretical framework for reconstructing the helicity quantum initial state of a general scattering process.
arXiv Detail & Related papers (2023-10-16T21:23:42Z) - GRAPE optimization for open quantum systems with time-dependent
decoherence rates driven by coherent and incoherent controls [77.34726150561087]
The GRadient Ascent Pulse Engineering (GRAPE) method is widely used for optimization in quantum control.
We adopt GRAPE method for optimizing objective functionals for open quantum systems driven by both coherent and incoherent controls.
The efficiency of the algorithm is demonstrated through numerical simulations for the state-to-state transition problem.
arXiv Detail & Related papers (2023-07-17T13:37:18Z) - Variational quantum algorithm for generalized eigenvalue problems and
its application to the finite element method [2.957189619293782]
Generalized eigenvalue problems (GEPs) play an important role in the variety of fields including engineering, machine learning and quantum chemistry.
This paper aims at extending sequential quantum sequentials for GEPs.
arXiv Detail & Related papers (2023-02-24T12:39:27Z) - Quantum Worst-Case to Average-Case Reductions for All Linear Problems [66.65497337069792]
We study the problem of designing worst-case to average-case reductions for quantum algorithms.
We provide an explicit and efficient transformation of quantum algorithms that are only correct on a small fraction of their inputs into ones that are correct on all inputs.
arXiv Detail & Related papers (2022-12-06T22:01:49Z) - The complexity of quantum support vector machines [1.7887848708497243]
Quantum support vector machines employ quantum circuits to define the kernel function.
We show that the dual problem can be solved in $O(M4.67/varepsilon2)$ quantum circuit evaluations.
arXiv Detail & Related papers (2022-02-28T19:01:17Z) - A theory of quantum subspace diagonalization [3.248953303528541]
We show that a quantum subspace diagonalization algorithm can accurately compute the smallest eigenvalue of a large Hermitian matrix.
Our results can be of independent interest to solving eigenvalue problems outside the context of quantum computation.
arXiv Detail & Related papers (2021-10-14T16:09:07Z) - Sampling electronic structure QUBOs with Ocean and Mukai solvers [44.62475518267084]
The most advanced D-Wave Advantage quantum annealer has 5000+ qubits, however, every qubit is connected to a small number of neighbors.
To compensate for the reduced number of qubits, one has to rely on special software such as qbsolv.
We find that the Mukai QUBO solver outperforms the Ocean qbsolv for all calculations done in the present work.
arXiv Detail & Related papers (2021-02-01T23:16:42Z) - 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.