The Importance of the Spectral Gap in Estimating Ground-State Energies

Authors Abhinav Deshpande , Alexey V. Gorshkov , Bill Fefferman



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Author Details

Abhinav Deshpande
  • Joint Center for Quantum Information and Computer Science and Joint Quantum Institute, NIST/University of Maryland, College Park, MD, USA
  • Institute for Quantum Information and Matter, California Institute of Technology, Pasadena, CA, USA
Alexey V. Gorshkov
  • Joint Center for Quantum Information and Computer Science and Joint Quantum Institute, NIST/University of Maryland, College Park, MD, USA
Bill Fefferman
  • Department of Computer Science, University of Chicago, IL, USA

Acknowledgements

We thank Sevag Gharibian, Hosho Katsura, Cedric Lin, Yupan Liu, Zachary Remscrim, and an anonymous referee for useful feedback on the manuscript and Peter Love, Yuan Su, Minh Tran, and Seth Whitsitt for helpful discussions.

Cite AsGet BibTex

Abhinav Deshpande, Alexey V. Gorshkov, and Bill Fefferman. The Importance of the Spectral Gap in Estimating Ground-State Energies. In 13th Innovations in Theoretical Computer Science Conference (ITCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 215, pp. 54:1-54:6, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
https://doi.org/10.4230/LIPIcs.ITCS.2022.54

Abstract

The field of quantum Hamiltonian complexity lies at the intersection of quantum many-body physics and computational complexity theory, with deep implications to both fields. The main object of study is the Local Hamiltonian problem, which is concerned with estimating the ground-state energy of a local Hamiltonian and is complete for the class QMA, a quantum generalization of the class NP. A major challenge in the field is to understand the complexity of the Local Hamiltonian problem in more physically natural parameter regimes. One crucial parameter in understanding the ground space of any Hamiltonian in many-body physics is the spectral gap, which is the difference between the smallest two eigenvalues. Despite its importance in quantum many-body physics, the role played by the spectral gap in the complexity of the Local Hamiltonian problem is less well-understood. In this work, we make progress on this question by considering the precise regime, in which one estimates the ground-state energy to within inverse exponential precision. Computing ground-state energies precisely is a task that is important for quantum chemistry and quantum many-body physics. In the setting of inverse-exponential precision (promise gap), there is a surprising result that the complexity of Local Hamiltonian is magnified from QMA to PSPACE, the class of problems solvable in polynomial space (but possibly exponential time). We clarify the reason behind this boost in complexity. Specifically, we show that the full complexity of the high precision case only comes about when the spectral gap is exponentially small. As a consequence of the proof techniques developed to show our results, we uncover important implications for the representability and circuit complexity of ground states of local Hamiltonians, the theory of uniqueness of quantum witnesses, and techniques for the amplification of quantum witnesses in the presence of postselection.

Subject Classification

ACM Subject Classification
  • Theory of computation → Quantum complexity theory
  • Theory of computation → Complexity classes
  • Theory of computation → Problems, reductions and completeness
Keywords
  • Local Hamiltonian problem
  • PSPACE
  • PP
  • QMA

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