Quantum subspace diagonalization methods are an exciting new class of algorithms for solving large scale eigenvalue problems using quantum computers. Unfortunately, these methods require the solution of an ill-conditioned generalized eigenvalue problem, with a matrix pencil corrupted by a non-negligible amount of noise that is far above the machine precision. Despite pessimistic predictions from classical perturbation theories, these methods can perform reliably well if the generalized eigenvalue problem is solved using a standard truncation strategy. We provide a theoretical analysis of this surprising phenomenon, proving that under certain natural conditions, a quantum subspace diagonalization algorithm can accurately compute the smallest eigenvalue of a large Hermitian matrix. We give numerical experiments demonstrating the effectiveness of the theory and providing practical guidance for the choice of truncation level.
We derive new uniqueness results for pLr, Lr, 1q-type block-term decompositions of third-order tensors by drawing connections to sparse component analysis (SCA). It is shown that our uniqueness results have a natural application in the context of the blind source separation problem, since they ensure uniqueness even amongst pLr, Lr, 1q-decompositions with incomparable rank profiles, allowing for stronger separation results for signals consisting of sums of exponentials in the presence of common poles among the source signals. As a byproduct, this line of ideas also suggests a new approach for computing pLr, Lr, 1q-decompositions, which proceeds by sequentially computing a canonical polyadic decomposition (CPD) of the input tensor, followed by performing a sparse factorization on the third factor matrix.
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