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 nonnegligible amount of noise that is far above the machine precision. Despite pessimistic predictions from classical worst-case perturbation theories, these methods can perform reliably well if the generalized eigenvalue problem is solved using a standard truncation strategy. By leveraging and advancing classical results in matrix perturbation theory, 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. Our new results can also be of independent interest to solving eigenvalue problems outside the context of quantum computation.


  1. quantum subspace diagonalization
  2. quantum linear algebra
  3. generalized eigenvalue problem
  4. matrix perturbation theory

MSC codes

  1. 68Q12
  2. 65F15
  3. 15A22
  4. 15A45

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Supplementary Material

PLEASE NOTE: These supplementary files have not been peer-reviewed.

Index of Supplementary Materials

Title of paper: A Theory of Quantum Subspace Diagonalization

Authors: Ethan N. Epperly, Lin Lin, and Yuji Nakatsukasa

File: supplement.pdf

Type: PDF

Contents: Additional proofs and numerical experiments.


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Information & Authors


Published In

cover image SIAM Journal on Matrix Analysis and Applications
SIAM Journal on Matrix Analysis and Applications
Pages: 1263 - 1290
ISSN (online): 1095-7162


Submitted: 15 November 2021
Accepted: 22 April 2022
Published online: 1 August 2022


  1. quantum subspace diagonalization
  2. quantum linear algebra
  3. generalized eigenvalue problem
  4. matrix perturbation theory

MSC codes

  1. 68Q12
  2. 65F15
  3. 15A22
  4. 15A45



Funding Information

Office of Multidisciplinary Activities https://doi.org/10.13039/100006091 : OMA-2016245
Simons Foundation https://doi.org/10.13039/100000893
U.S. Department of Energy https://doi.org/10.13039/100000015 : DE-SC0021110, DE-SC0017867

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