A Theory of Quantum Subspace Diagonalization
Abstract
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 illconditioned generalized eigenvalue problem, with a matrix pencil corrupted by a nonnegligible 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.
 Publication:

arXiv eprints
 Pub Date:
 October 2021
 arXiv:
 arXiv:2110.07492
 Bibcode:
 2021arXiv211007492E
 Keywords:

 Quantum Physics;
 Mathematics  Numerical Analysis;
 68Q12;
 65F15;
 15A22;
 15A45
 EPrint:
 42 pages, 13 figures