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Improved Approximations for Extremal Eigenvalues of Sparse Hamiltonians

Authors Daniel Hothem , Ojas Parekh , Kevin Thompson

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

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Abstract

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.

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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) https://doi.org/10.4230/LIPIcs.TQC.2023.6

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

Funding

This material is based upon work supported by the U.S. Department of Energy, Office of Science, Office of Advanced Scientific Computing Research, National Quantum Information Science Research Centers, Exploratory Research for Extreme Scale Science program. Support is also acknowledged from the Accelerated Research in Quantum Computing program under the same office.

Acknowledgements

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 https://www.energy.gov/downloads/doe-public-access-plan.

References

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