Improved Hardness Results for the Guided Local Hamiltonian Problem

Authors Chris Cade, Marten Folkertsma, Sevag Gharibian, Ryu Hayakawa, François Le Gall, Tomoyuki Morimae, Jordi Weggemans

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Author Details

Chris Cade
  • QuSoft and University of Amsterdam (UvA), The Netherlands
Marten Folkertsma
  • QuSoft and CWI, Amsterdam, The Netherlands
Sevag Gharibian
  • Paderborn Universität, Germany
Ryu Hayakawa
  • Kyoto University, Japan
François Le Gall
  • Nagoya University, Japan
Tomoyuki Morimae
  • Kyoto University, Japan
Jordi Weggemans
  • QuSoft and CWI, Amsterdam, The Netherlands


We thank Jonas Helsen for feedback on an earlier draft, and Ronald de Wolf for helpful comments.

Cite AsGet BibTex

Chris Cade, Marten Folkertsma, Sevag Gharibian, Ryu Hayakawa, François Le Gall, Tomoyuki Morimae, and Jordi Weggemans. Improved Hardness Results for the Guided Local Hamiltonian Problem. In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 32:1-32:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


Estimating the ground state energy of a local Hamiltonian is a central problem in quantum chemistry. In order to further investigate its complexity and the potential of quantum algorithms for quantum chemistry, Gharibian and Le Gall (STOC 2022) recently introduced the guided local Hamiltonian problem (GLH), which is a variant of the local Hamiltonian problem where an approximation of a ground state (which is called a guiding state) is given as an additional input. Gharibian and Le Gall showed quantum advantage (more precisely, BQP-completeness) for GLH with 6-local Hamiltonians when the guiding state has fidelity (inverse-polynomially) close to 1/2 with a ground state. In this paper, we optimally improve both the locality and the fidelity parameter: we show that the BQP-completeness persists even with 2-local Hamiltonians, and even when the guiding state has fidelity (inverse-polynomially) close to 1 with a ground state. Moreover, we show that the BQP-completeness also holds for 2-local physically motivated Hamiltonians on a 2D square lattice or a 2D triangular lattice. Beyond the hardness of estimating the ground state energy, we also show BQP-hardness persists when considering estimating energies of excited states of these Hamiltonians instead. Those make further steps towards establishing practical quantum advantage in quantum chemistry.

Subject Classification

ACM Subject Classification
  • Theory of computation → Quantum complexity theory
  • Quantum computing
  • Quantum advantage
  • Quantum Chemistry
  • Guided Local Hamiltonian Problem


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