Short proof of a spectral Chernoff bound for local Hamiltonians

Abstract

We give a simple proof of a Chernoff bound for the spectrum of a $k$-local Hamiltonian based on Weyl's inequalities. The complexity of estimating the spectrum's $\epsilon (n)$-th quantile up to constant relative error thus exhibits the following dichotomy: For $\epsilon (n)=d^{-n}$ the problem is NP-hard and maybe even QMA-hard, yet there exists constant $a>1$ such that the problem is trivial for $\epsilon (n)=a^{-n}$. We note that a related Chernoff bound due to Kuwahara and Saito (Ann. Phys. '20) for a generalized problem is also sufficient to establish such a dichotomy, its proof relying on a careful analysis of the \emph{cluster expansion}.

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