Many quantum algorithms are developed to evaluate eigenvalues for Hermitian matrices. However, few practical approach exists for the eigenanalysis of nonHermintian ones, such as arising from modern power systems. The main difficulty lies in the fact that, as the eigenvector matrix of a general matrix can be nonunitary, solving a general eigenvalue problem is inherently incompatible with existing unitarygatebased quantum methods. To fill this gap, this paper introduces a Variational Quantum Universal Eigensolver (VQUE), which is deployable on noisy intermediate scale quantum computers. Our new contributions include: (1) The first universal variational quantum algorithm capable of evaluating the eigenvalues of nonHermitian matrices—Inspired by Schur’s triangularization theory, VQUE unitarizes the eigenvalue problem to a procedure of searching unitary transformation matrices via quantum devices; (2) A Quantum Process Snapshot technique is devised to make VQUE maintain the potential quantum advantage inherited from the original variational quantum eigensolver—With additional
A Fast and Effective MemristorBased Method for Finding Approximate Eigenvalues and Eigenvectors of Nonnegative Matrices
Throughout many scientific and engineering fields, including control theory, quantum mechanics, advanced dynamics, and network theory, a great many important applications rely on the spectral decomposition of matrices. Traditional methods such as the power iteration method, Jacobi eigenvalue method, and QR decomposition are commonly used to compute the eigenvalues and eigenvectors of a square and symmetric matrix. However, these methods suffer from certain drawbacks: in particular, the power iteration method can only find the leading eigenpair (i.e., the largest eigenvalue and its corresponding eigenvector), while the Jacobi and QR decomposition methods face significant performance limitations when facing with large scale matrices. Typically, even producing approximate eigenpairs of a general square matrix requires at least O(N^3) time complexity, where N is the number of rows of the matrix. In this work, we exploit the newly developed memristor technology to propose a lowcomplexity, scalable memristorbased method for deriving a set of dominant eigenvalues and eigenvectors for real symmetric nonnegative matrices. The time complexity for our proposed algorithm is O(N^2 /Δ) (where Δ governs the accuracy). We present experimental studies to simulate the memristorsupporting algorithm, with results demonstrating that the average error for our method is within 4%, while its performance is up to 1.78X better than traditional methods.
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 Award ID(s):
 1637559
 NSFPAR ID:
 10110063
 Date Published:
 Journal Name:
 2018 IEEE Computer Society Annual Symposium on VLSI (ISVLSI)
 Page Range / eLocation ID:
 563 to 568
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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