The commuting local Hamiltonian problem on locally expanding graphs is approximable in (formula presented)

The local Hamiltonian problem is famously complete for the class Aaronson, S.: The quantum (formula presented) manifesto (2006), the quantum analogue of Aaronson, S.: The quantum (formula presented) manifesto (2006). The complexity of its semiclassical version, in which the terms of the Hamiltonian are required to commute (the Aaronson, S.: The quantum (formula presented) manifesto (2006) problem), has attracted considerable attention recently due to its intriguing nature, as well as in relation to growing interest in the Aaronson, S.: The quantum (formula presented) manifesto (2006) conjecture. We show here that if the underlying bipartite interaction graph of the Aaronson, S.: The quantum (formula presented) manifesto (2006) instance is a good locally expanding graph, namely the expansion of any constant-size set is Aaronson, S.: The quantum (formula presented) manifesto (2006)-close to optimal, then approximating its ground energy to within additive factor Aaronson, S.: The quantum (formula presented) manifesto (2006) lies in Aaronson, S.: The quantum (formula presented) manifesto (2006). The proof holds for Aaronson, S.: The quantum (formula presented) manifesto (2006)-local Hamiltonians for any constant Aaronson, S.: The quantum (formula presented) manifesto (2006) and any constant dimensionality of particles Aaronson, S.: The quantum (formula presented) manifesto (2006). We also show that the approximation problem of Aaronson, S.: The quantum (formula presented) manifesto (2006) on such good local expanders is Aaronson, S.: The quantum (formula presented) manifesto (2006)-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 Aaronson, S.: The quantum (formula presented) manifesto (2006) proof) in an attempt to prove the quantum Aaronson, S.: The quantum (formula presented) manifesto (2006) conjecture. A related result was discovered recently independently by Brandão and Harrow, for Aaronson, S.: The quantum (formula presented) manifesto (2006)-local general Hamiltonians, bounding the quantum hardness of the approximation problem on good expanders, though no Aaronson, S.: The quantum (formula presented) manifesto (2006) hardness is known in that case.