The local Hamiltonian problem is famously complete for the class QMA\documentclass[12pt]{minimal}
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\begin{document}$$\mathsf{{QMA}}$$\end{document}, the quantum analogue of NP\documentclass[12pt]{minimal}
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\begin{document}$$\mathsf{{NP}}$$\end{document}. The complexity of its semiclassical version, in which the terms of the Hamiltonian are required to commute (the CLH\documentclass[12pt]{minimal}
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\begin{document}$$\mathsf{{CLH}}$$\end{document} problem), has attracted considerable attention recently due to its intriguing nature, as well as in relation to growing interest in the qPCP\documentclass[12pt]{minimal}
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\begin{document}$$\mathsf{{qPCP}}$$\end{document} conjecture. We show here that if the underlying bipartite interaction graph of the CLH\documentclass[12pt]{minimal}
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\begin{document}$$\mathsf{{CLH}}$$\end{document} instance is a good locally expanding graph, namely the expansion of any constant-size set is ε\documentclass[12pt]{minimal}
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\begin{document}$$\varepsilon $$\end{document}-close to optimal, then approximating its ground energy to within additive factor O(ε)\documentclass[12pt]{minimal}
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\begin{document}$$O(\varepsilon )$$\end{document} lies in NP\documentclass[12pt]{minimal}
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\begin{document}$$\mathsf{{NP}}$$\end{document}. The proof holds for k\documentclass[12pt]{minimal}
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\begin{document}$$k$$\end{document}-local Hamiltonians for any constant k\documentclass[12pt]{minimal}
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\begin{document}$$k$$\end{document} and any constant dimensionality of particles d\documentclass[12pt]{minimal}
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\begin{document}$$d$$\end{document}. We also show that the approximation problem of CLH\documentclass[12pt]{minimal}
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\begin{document}$$\mathsf{{CLH}}$$\end{document} on such good local expanders is NP\documentclass[12pt]{minimal}
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\begin{document}$$\mathsf{{NP}}$$\end{document}-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\documentclass[12pt]{minimal}
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\begin{document}$$\mathsf{{PCP}}$$\end{document} proof) in an attempt to prove the quantum PCP\documentclass[12pt]{minimal}
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\begin{document}$$\mathsf{{PCP}}$$\end{document} conjecture. A related result was discovered recently independently by Brandão and Harrow, for 2\documentclass[12pt]{minimal}
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\begin{document}$$2$$\end{document}-local general Hamiltonians, bounding the quantum hardness of the approximation problem on good expanders, though no NP\documentclass[12pt]{minimal}
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\begin{document}$$\mathsf{{NP}}$$\end{document} hardness is known in that case.
