The commuting local Hamiltonian problem on locally expanding graphs is approximable in NP.

Dorit Aharonov, Lior Eldar*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

12 Scopus citations


The local Hamiltonian problem is famously complete for the class QMA, the quantum analogue of NP. The complexity of its semiclassical version, in which the terms of the Hamiltonian are required to commute (the CLH problem), has attracted considerable attention recently due to its intriguing nature, as well as in relation to growing interest in the qPCP conjecture. We show here that if the underlying bipartite interaction graph of the CLH instance is a good locally expanding graph, namely the expansion of any constant-size set is ε-close to optimal, then approximating its ground energy to within additive factor O(ε) lies in NP. The proof holds for k-local Hamiltonians for any constant k and any constant dimensionality of particles d. We also show that the approximation problem of CLH on such good local expanders is NP-hard. This implies that too good local expansion of the interaction graph constitutes an obstacle against quantum hardness of the approximation problem, though it retains its classical hardness. The result highlights new difficulties in trying to mimic classical proofs (in particular, Dinur’s PCP proof) in an attempt to prove the quantum PCP conjecture. A related result was discovered recently independently by Brandão and Harrow, for 2-local general Hamiltonians, bounding the quantum hardness of the approximation problem on good expanders, though no NP hardness is known in that case.
Original languageEnglish
Article number1
Pages (from-to)83-101
Number of pages19
JournalQuantum Information Processing
Issue number1
StatePublished - 26 Nov 2014

Bibliographical note

Funding Information:
The authors would like to thank Irit Dinur and Gil Kalai for insightful discussions and Prasad Raghavendra, Luca Trevisan and Umesh Vazirani for helpful comments. The research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013)/ERC Grant Agreement No. 280157, and from ISF Grant No. 1446/09.

Publisher Copyright:
© 2014, Springer Science+Business Media New York.


  • Approximation
  • Commuting local Hamiltonian
  • Local Hamiltonian
  • PCP
  • Quantum PCP


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