Improved Approximations for Extremal Eigenvalues of Sparse Hamiltonians

Authors Daniel Hothem , Ojas Parekh , Kevin Thompson

Thumbnail PDF


  • Filesize: 0.63 MB
  • 10 pages

Document Identifiers

Author Details

Daniel Hothem
  • Quantum Algorithms and Applications Collaboratory, Sandial National Laboratories, Livermore, CA, USA
Ojas Parekh
  • Quantum Algorithms and Applications Collaboratory, Sandial National Laboratories, Albuquerque, NM, USA
Kevin Thompson
  • Quantum Algorithms and Applications Collaboratory, Sandial National Laboratories, Albuquerque, NM, USA


We thank Yaroslav Herasymenko for an insightful contribution to Lemma 8. This article has been authored by an employee of National Technology & Engineering Solutions of Sandia, LLC under Contract No. DE-NA0003525 with the U.S. Department of Energy (DOE). The employee owns all right, title and interest in and to the article and is solely responsible for its contents. The United States Government retains and the publisher, by accepting the article for publication, acknowledges that the United States Government retains a non-exclusive, paid-up, irrevocable, world-wide license to publish or reproduce the published form of this article or allow others to do so, for United States Government purposes. The DOE will provide public access to these results of federally sponsored research in accordance with the DOE Public Access Plan

Cite AsGet BibTex

Daniel Hothem, Ojas Parekh, and Kevin Thompson. Improved Approximations for Extremal Eigenvalues of Sparse Hamiltonians. In 18th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 266, pp. 6:1-6:10, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


We give a classical 1/(qk+1)-approximation for the maximum eigenvalue of a k-sparse fermionic Hamiltonian with strictly q-local terms, as well as a 1/(4k+1)-approximation when the Hamiltonian has both 2-local and 4-local terms. More generally we obtain a 1/O(qk²)-approximation for k-sparse fermionic Hamiltonians with terms of locality at most q. Our techniques also yield analogous approximations for k-sparse, q-local qubit Hamiltonians with small hidden constants and improved dependence on q.

Subject Classification

ACM Subject Classification
  • Theory of computation → Approximation algorithms analysis
  • Mathematics of computing → Approximation algorithms
  • Approximation algorithms
  • Extremal eigenvalues
  • Sparse Hamiltonians
  • Fermionic Hamiltonians
  • Qubit Hamiltonians


  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    PDF Downloads


  1. Sergey Bravyi. Lagrangian representation for fermionic linear optics. Quantum Information & Computation., 5(3):216-238, 2005. URL:
  2. Sergey Bravyi, David Gosset, Robert Koenig, and Kristan Temme. Approximation algorithms for quantum many-body problems. Journal of Mathematical Physics, 60, 2019. URL:
  3. Aram W. Harrow and Ashley Montanaro. Extremal eigenvalues of local Hamiltonians. Quantum, 1:6, 2017. URL:
  4. Matthew B. Hastings and Ryan O'Donnell. Optimizing strongly interacting fermionic hamiltonians. In Proceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2022, pages 776-789, New York, NY, USA, 2022. Association for Computing Machinery. URL:
  5. Yaroslav Herasymenko, Maarten Stroeks, Jonas Helsen, and Barbara Terhal. Optimizing sparse fermionic hamiltonians, 2022. URL:
  6. John Hubbard. Electron correlations in narrow energy bands. Proceedings of the Royal Society R. Society A, 276:238-257, 1963. URL:
  7. Julia Kempe, Alexei Kitaev, and Oded Regev. The complexity of the local hamiltonian problem. In Kamal Lodaya and Meena Mahajan, editors, FSTTCS 2004: Foundations of Software Technology and Theoretical Computer Science, pages 372-383, Berlin, Heidelberg, 2005. Springer Berlin Heidelberg. URL:
  8. Alexei Yu Kitaev, Alexander Shen, Mikhail N Vyalyi, and Mikhail N Vyalyi. Classical and quantum computation. Number 47 in Graduate Studies in Mathematics. American Mathematical Society, 2002. URL:
Questions / Remarks / Feedback

Feedback for Dagstuhl Publishing

Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail