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Cited by 865Quantum many-body systemsQuantum Computing Algorithms and ArchitectureNumerical methods for differential equations

Time-Dependent Variational Principle for Quantum Lattices

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Haegeman, Jutho, Cirac, J. Ignacio, Osborne, Tobias J., Pižorn, Iztok, Verschelde, Henri, Verstraete, Frank·2011·Physical Review Letters
DOI 10.1103/physrevlett.107.070601
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Abstract

We develop a new algorithm based on the time-dependent variational principle applied to matrix product states to efficiently simulate the real- and imaginary-time dynamics for infinite one-dimensional quantum lattices. This procedure (i) is argued to be optimal, (ii) does not rely on the Trotter decomposition and thus has no Trotter error, (iii) preserves all symmetries and conservation laws, and (iv) has low computational complexity. The algorithm is illustrated by using both an imaginary-time and a real-time example.

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Crossref · 10.1103/physrevlett.107.070601

Cites

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  • Time-evolving an MPS with χ = 1 with TDVP is ill-defined, as many long-range interactions are lost during the projection step and the resulting time evolution would be wrong.

    “Time-evolving an MPS with χ = 1 with the time-dependent variational principle (TDVP, see next subsection for more details) is ill-defined, as many of the long-range interactions are lost during the projection step and the resulting time evolution would be wrong 51,76,77 .”

    From Beyond-classical computation in quantum simulation (2025) · cites this paper as [76]

    The discussion accompanying Fig. 1 describes TDVP as projecting the time-evolution direction onto the MPS tangent plane, supporting the reference to a projection step. Our copy does not say that bond dimension χ = 1 makes TDVP ill-defined, that long-range interactions are lost in that case, or that the resulting evolution is wrong; it instead describes the projection as the best approximation within the chosen manifold. (LLM, conf=0.82)

    Experimental — accuracy not yet measured

  • TDVP projects the right-hand side of the Schrödinger equation onto the tangent space so that the evolution never leaves the manifold.

    “The key ingredient is to project the right-hand side of the Schrödinger equation, Ĥ | , onto the tangent space, so that the evolution never leaves the manifold.”

    From Unifying time evolution and optimization with matrix product states (2016) · cites this paper as [18]

  • The original TDVP implementation depends on inverses of matrices conditioned by Schmidt coefficients, similar to the original TEBD implementation.

    “The TDVP equations define a simultaneous update for all MPS tensors and, as a consequence thereof, its original implementation depends on inverses of matrices conditioned by the Schmidt coefficients, similar to the original TEBD implementation.”

    From Unifying time evolution and optimization with matrix product states (2016) · cites this paper as [18]

  • The Dirac-Frenkel time-dependent variational principle was formulated only recently for the variational class of matrix product states.

    “An alternative idea for time evolution, the Dirac-Frenkel time-dependent variational principle (TDVP), was formulated only recently for the variational class of MPS [18,19].”

    From Unifying time evolution and optimization with matrix product states (2016) · cites this paper as [18]

  • The algorithm in Ref. [18] used explicit Euler integration to update all parameters simultaneously, which required explicit inverses of matrices containing very small singular values.

    “The algorithm presented in Ref. [18] used the explicit Euler integration scheme to simultaneously update all parameters, which was only possible by using explicit inverses of matrices containing very small singular values.”

    From Unifying time evolution and optimization with matrix product states (2016) · cites this paper as [18]

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Cites

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Research works printed but not linked: 19

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  11. [13]13. F. Verstraete, J. J. Garcia-Ripoll, and J. I. Cirac, Phys. Rev. Lett. 93, 207204 (2004).
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Authors
Haegeman, J., Cirac, J. I., Osborne, T. J., Pižorn, I., Verschelde, H., & Verstraete, F.
Affiliations
  • Ghent University (BE)
  • Max Planck Institute of Quantum Optics (DE)
  • Leibniz University Hannover (DE)
  • University of Vienna (AT)
ORCID
Venue
Physical Review Letters
Publisher
American Physical Society (APS)
Published
2011-08-10
Volume
107
Issue
7
Pages
070601-070601
ISSN
  • 0031-9007 (ISSN-L)
  • 1079-7114
DOI
10.1103/physrevlett.107.070601
OpenAlex
W1974645033
Cited by
865
References
24
Citations by year
2026: 84 · 2025: 142 · 2024: 92 · 2023: 72 · 2022: 83 · 2021: 90 · 2020: 92 · 2019: 54 · 2018: 49 · 2017: 30 · 2016: 14 · 2015: 24 · 2014: 16 · 2013: 15 · 2012: 8
FWCI
6.76
Type
journal article
Language
en
Open access
Licence
link.aps.org/licenses/aps-default-license (vor)
Locations
  • Physical Review Letters (publishedVersion)
  • PubMed (publishedVersion)
  • arXiv (Cornell University) (submittedVersion)
  • Ghent University Academic Bibliography (Ghent University) (submittedVersion, other-oa)
  • Institutional Repository of Leibniz Universität Hannover (Leibniz Universität Hannover) (submittedVersion, cc-by-nc-nd)
  • Max Planck Institute for Plasma Physics (submittedVersion)
  • MPG.PuRe (Max Planck Society) (submittedVersion)
  • arXiv (Cornell University)
  • Leibniz Universität Hannover
Funders
  • Deutsche Forschungsgemeinschaft
  • Fonds Wetenschappelijk Onderzoek
  • Vlaamse regering
Topics
  • Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum many-body systems
  • Physical Sciences › Computer Science › Artificial Intelligence › Quantum Computing Algorithms and Architecture
  • Physical Sciences › Mathematics › Numerical Analysis › Numerical methods for differential equations
Keywords
time-dependent variational principle, matrix product states, real-time dynamics, Trotter error, symmetry preservation, conservation laws, variational quantum algorithms

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