
Randomized block Krylov methods for approximating extreme eigenvalues
Randomized block Krylov subspace methods form a powerful class of algori...
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Randomized Linear Algebra Approaches to Estimate the Von Neumann Entropy of Density Matrices
The von Neumann entropy, named after John von Neumann, is the extension ...
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Spectral sparsification of matrix inputs as a preprocessing step for quantum algorithms
We study the potential utility of classical techniques of spectral spars...
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A remark on approximating permanents of positive definite matrices
Let A be an n × n positive definite Hermitian matrix with all eigenvalue...
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Randomized block Krylov space methods for trace and logdeterminant estimators
We present randomized algorithms based on block Krylov space method for ...
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Approximate Graph Spectral Decomposition with the Variational Quantum Eigensolver
Spectral graph theory is a branch of mathematics that studies the relati...
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Reduced Sum Implementation of the BURA Method for Spectral Fractional Diffusion Problems
The numerical solution of spectral fractional diffusion problems in the ...
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Quantum algorithms for spectral sums
We propose and analyze new quantum algorithms for estimating the most common spectral sums of symmetric positive definite (SPD) matrices. For a function f and a matrix A ∈ℝ^n× n, the spectral sum is defined as S_f(A) :=Tr[f(A)] = ∑_j f(λ_j), where λ_j are the eigenvalues. Examples of spectral sums are the von Neumann entropy, the trace of inverse, the logdeterminant, and the Schattenp norm, where the latter does not require the matrix to be SPD. The fastest classical randomized algorithms estimate these quantities have a runtime that depends at least linearly on the number of nonzero components of the matrix. Assuming quantum access to the matrix, our algorithms are sublinear in the matrix size, and depend at most quadratically on other quantities, like the condition number and the approximation error, and thus can compete with most of the randomized and distributed classical algorithms proposed in recent literature. These algorithms can be used as subroutines for solving many practical problems, for which the estimation of a spectral sum often represents a computational bottleneck.
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