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 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.

Keywords

  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


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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.

References

1.
R. Bhatia, Matrix Analysis, Grad. Texts in Math. 169, Springer-Verlag, New York, 1997, https://doi.org/10.1007/978-1-4612-0653-8.
2.
\AA. Björck, Numerical Methods in Matrix Computations, Texts in Appl. Math. 59, Springer International, New York, 2015, https://doi.org/10.1007/978-3-319-05089-8.
3.
Y. Cai, Z. Bai, J. E. Pask, and N. Sukumar, Hybrid preconditioning for iterative diagonalization of ill-conditioned generalized eigenvalue problems in electronic structure calculations, J. Comput. Phys., 255 (2013), pp. 16--30.
4.
A. M. Childs, Y. Su, M. C. Tran, N. Wiebe, and S. Zhu, Theory of Trotter error with commutator scaling, Phys. Rev. X, 11 (2021), 011020, https://doi.org/10.1103/PhysRevX.11.011020.
5.
J. I. Colless, V. V. Ramasesh, D. Dahlen, M. S. Blok, M. E. Kimchi-Schwartz, J. R. McClean, J. Carter, W. A. de Jong, and I. Siddiqi, Computation of molecular spectra on a quantum processor with an error-resilient algorithm, Phys. Rev. X, 8 (2018), 011021, https://doi.org/10.1103/PhysRevX.8.011021.
6.
C. L. Cortes and S. K. Gray, Quantum Krylov Subspace Algorithms for Ground and Excited State Energy Estimation, https://arxiv.org/abs/2109.06868, 2021.
7.
C. Davis and W. M. Kahan, The rotation of eigenvectors by a perturbation. III, SIAM J. Numer. Anal., 7 (1970), pp. 1--46, https://doi.org/10.1137/0707001.
8.
P. Drineas and I. C. F. Ipsen, Low-rank matrix approximations do not need a singular value gap, SIAM J. Matrix Anal. Appl., 40 (2019), pp. 299--319, https://doi.org/10.1137/18M1163658.
9.
A. Emami-Naeini and P. Van Dooren, Computation of zeros of linear multivariable systems, Automatica, 18 (1982), pp. 415--430, https://doi.org/10.1016/0005-1098(82)90070-X.
10.
G. Fix and R. Heiberger, An algorithm for the ill-conditioned generalized eigenvalue problem, SIAM J. Numer. Anal., 9 (1972), pp. 78--88, https://doi.org/10.1137/0709009.
11.
G. H. Golub and C. F. Van Loan, Matrix Computations, 4th ed., Johns Hopkins University Press, Baltimore, 2013.
12.
L. A. Gribov and B. K. Novosadov, Use of overcomplete basis sets in quantum-chemical calculations, J. Molecular Structure THEOCHEM, 136 (1986), pp. 387--389, https://doi.org/10.1016/0166-1280(86)80152-X.
13.
W. J. Huggins, J. Lee, U. Baek, B. O'Gorman, and K. B. Whaley, A non-orthogonal variational quantum eigensolver, New J. Phys., 22 (2020), 073009, https://doi.org/10.1088/1367-2630/ab867b.
14.
M. Jungen and K. Kaufmann, The Fix--Heiberger procedure for solving the generalized ill-conditioned symmetric eigenvalue problem, Int. J. Quantum Chem., 41 (1992), pp. 387--397.
15.
K. Klymko, C. Mejuto-Zaera, S. J. Cotton, F. Wudarski, M. Urbanek, D. Hait, M. Head-Gordon, K. B. Whaley, J. Moussa, N. Wiebe, W. A. de Jong, and N. M. Tubman, Real Time Evolution for Ultracompact Hamiltonian Eigenstates on Quantum Hardware, https://arxiv.org/abs/2103.08563, 2021.
16.
M. Lotz and V. Noferini, Wilkinson's bus: Weak condition numbers, with an application to singular polynomial eigenproblems, Found. Comput. Math., 20 (2020), pp. 1439--1473, https://doi.org/10.1007/s10208-020-09455-y.
17.
P.-O. Löwdin, Group algebra, convolution algebra, and applications to quantum mechanics, Rev. Modern Phys., 39 (1967), pp. 259--287, https://doi.org/10.1103/RevModPhys.39.259.
18.
V. A. Mandelshtam and H. S. Taylor, Harmonic inversion of time signals and its applications, J. Chem. Phys., 107 (1997), pp. 6756--6769, https://doi.org/10.1063/1.475324.
19.
R. Mathias and C.-K. Li, The Definite Generalized Eigenvalue Problem: A New Perturbation Theory, T-NAREP 457, Manchester Centre for Computational Mathematics, 2004.
20.
J. R. McClean, M. E. Kimchi-Schwartz, J. Carter, and W. A. de Jong, Hybrid quantum-classical hierarchy for mitigation of decoherence and determination of excited states, Phys. Rev. A, 95 (2017), 042308, https://doi.org/10.1103/PhysRevA.95.042308.
21.
M. Motta, C. Sun, A. T. K. Tan, M. J. O'Rourke, E. Ye, A. J. Minnich, F. G. S. L. Branda͂o, and G. K.-L. Chan, Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution, Nature Physics, 16 (2020), pp. 205--210, https://doi.org/10.1038/s41567-019-0704-4.
22.
G. Nannicini, An introduction to quantum computing, without the physics, SIAM Rev., 62 (2020), pp. 936--981.
23.
J. W. Negele and H. Orland, Quantum Many-Particle Systems, Westview, Boulder, CO, 1988.
24.
M. A. Nielsen and I. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, Cambridge, UK, 2000.
25.
B. N. Parlett, The Symmetric Eigenvalue Problem, Classics in Appl. Math. 20, SIAM, Philadelphia, 1998, https://doi.org/10.1137/1.9781611971163.
26.
R. M. Parrish and P. L. McMahon, Quantum Filter Diagonalization: Quantum Eigendecomposition Without Full Quantum Phase Estimation, https://arxiv.org/abs/1909.08925, 2019.
27.
J. Preskill, Quantum computing in the NISQ era and beyond, Quantum, 2 (2018), p. 79.
28.
J. Preskill, Quantum Computing 40 Years Later, https://arxiv.org/abs/2106.10522, 2021.
29.
Y. Saad, On the rates of convergence of the Lanczos and the block-Lanczos methods, SIAM J. Numer. Anal., 17 (1980), pp. 687--706, https://doi.org/10.1137/0717059.
30.
I. Sabzevari, A. Mahajan, and S. Sharma, An accelerated linear method for optimizing non-linear wavefunctions in variational Monte Carlo, J. Chem. Phys., 152 (2020), 024111, https://doi.org/10.1063/1.5125803.
31.
K. Seki and S. Yunoki, Quantum power method by a superposition of time-evolved states, Phys. Rev. X Quantum, 2 (2021), 010333.
32.
N. H. Stair, R. Huang, and F. A. Evangelista, A multireference quantum Krylov algorithm for strongly correlated electrons, J. Chem. Theory Comput., 16 (2020), pp. 2236--2245, https://doi.org/10.1021/acs.jctc.9b01125.
33.
G. W. Stewart, Perturbation bounds for the definite generalized eigenvalue problem, Linear Algebra Appl., 23 (1979), pp. 69--85, https://doi.org/10.1016/0024-3795(79)90094-6.
34.
G. W. Stewart and J.-G. Sun, Matrix Perturbation Theory, Computer Science and Scientific Computing, Academic Press, New York, 1990.
35.
J. A. Tropp, An Introduction to Matrix Concentration Inequalities, Found. Trends Machine Learning 8, Now Publishers, Hanover, 2015, pp. 1--230.
36.
M. R. Wall and D. Neuhauser, Extraction, through filter-diagonalization, of general quantum eigenvalues or classical normal mode frequencies from a small number of residues or a short-time segment of a signal. I. Theory and application to a quantum-dynamics model, J. Chem. Phys., 102 (1995), pp. 8011--8022, https://doi.org/10.1063/1.468999.
37.
P. Weinberg and M. Bukov, QuSpin: A Python package for dynamics and exact diagonalisation of quantum many body systems Part I: Spin chains, SciPost Phys., 2 (2017).
38.
P. Weinberg and M. Bukov, QuSpin: A Python package for dynamics and exact diagonalisation of quantum many body systems. Part II: Bosons, fermions and higher spins, SciPost Phys., 7 (2019).
39.
J. H. Wilkinson, Kronecker's canonical form and the QZ algorithm, Linear Algebra Appl., 28 (1979), pp. 285--303, https://doi.org/10.1016/0024-3795(79)90140-X.

Information & Authors

Information

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

History

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

Keywords

  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

Authors

Affiliations

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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