Advertisement

Corrected One-Site Density Matrix Renormalization Group and Alternating Minimal Energy Algorithm

  • Sergey V. Dolgov
  • Dmitry V. Savostyanov
Conference paper
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 103)

Abstract

Given in the title are two algorithms to compute the extreme eigenstate of a high-dimensional Hermitian matrix using the tensor train (TT)/matrix product states (MPS) representation. Both methods empower the traditional alternating direction scheme with the auxiliary (e.g. gradient) information, which substantially improves the convergence in many difficult cases. Being conceptually close, these methods have different derivation, implementation, theoretical and practical properties. We emphasize the differences, and reproduce the numerical example to compare the performance of two algorithms.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    S.V. Dolgov, D.V. Savostyanov, Alternating minimal energy methods for linear systems in higher dimensions. SIAM J. Sci. Comput. 36(5), A2248–A2271 (2014). doi: 10.1137/140953289CrossRefMathSciNetGoogle Scholar
  2. 2.
    M. Fannes, B. Nachtergaele, R. Werner, Finitely correlated states on quantum spin chains. Commun. Math. Phys. 144(3), 443–490 (1992)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    E. Jeckelmann, Dynamical density–matrix renormalization–group method. Phys. Rev. B 66, 045114 (2002)CrossRefGoogle Scholar
  4. 4.
    H. Munthe-Kaas, The convergence rate of inexact preconditioned steepest descent algorithm for solving linear systems, Numerical Analysis Report NA-87-04, Stanford University, 1987Google Scholar
  5. 5.
    I.V. Oseledets, Tensor-train decomposition. SIAM J. Sci. Comput. 33(5), 2295–2317 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    T. Rohwedder, A. Uschmajew, Local convergence of alternating schemes for optimization of convex problems in the TT format. SIAM J. Numer. Anal. 51(2), 1134–1162 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    U. Schollwöck, The density-matrix renormalization group in the age of matrix product states. Ann. Phys. 326(1), 96–192 (2011)CrossRefzbMATHGoogle Scholar
  8. 8.
    S.R. White, Density matrix formulation for quantum renormalization groups. Phys. Rev. Lett. 69(19), 2863–2866 (1992)CrossRefGoogle Scholar
  9. 9.
    ___ , Density matrix renormalization group algorithms with a single center site. Phys. Rev. B 72(18), 180403 (2005)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Max Planck Institute for Mathematics in the SciencesLeipzigGermany
  2. 2.School of ChemistryUniversity of SouthamptonSouthamptonUnited Kingdom

Personalised recommendations