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The Rotation-Invariant Hamiltonian Problem Is QMA_EXP-Complete

Authors Jon Nelson , Daniel Gottesman

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Subject Classification

ACM Subject Classification
  • Theory of computation → Quantum complexity theory
Keywords
  • Hamiltonian complexity
  • Local Hamiltonian problem
  • Monogamy of entanglement

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Abstract

In this work we study a variant of the local Hamiltonian problem where we restrict to Hamiltonians that live on a lattice and are invariant under translations and rotations of the lattice. In the one-dimensional case this problem is known to be QMA_EXP-complete. On the other hand, if we fix the lattice length then in the high-dimensional limit the ground state becomes unentangled due to arguments from mean-field theory. We take steps towards understanding this complexity spectrum by studying a problem that is intermediate between these two extremes. Namely, we consider the regime where the lattice dimension is arbitrary but fixed and the lattice length is scaled. We prove that this rotation-invariant Hamiltonian problem is QMA_EXP-complete answering an open question of [Gottesman and Irani, 2013]. This characterizes a broad parameter range in which these rotation-invariant Hamiltonians have high computational complexity.

Cite As Get BibTex

Jon Nelson and Daniel Gottesman. The Rotation-Invariant Hamiltonian Problem Is QMA_EXP-Complete. In 20th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 350, pp. 12:1-12:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.TQC.2025.12

Author Details

Jon Nelson
  • Joint Center for Quantum Information and Computer Science (QuICS), Department of Computer Science, University of Maryland, College Park, MD, USA
Daniel Gottesman
  • Joint Center for Quantum Information and Computer Science (QuICS), Department of Computer Science, University of Maryland, College Park, MD, USA

Funding

  • Nelson, Jon: supported by the National Science Foundation Graduate Research Fellowship Program under Grant No. DGE 2236417.

References

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