Short proof of a spectral Chernoff bound for local Hamiltonians

TL;DR

This paper presents a straightforward proof of a spectral Chernoff bound for local Hamiltonians, revealing a complexity dichotomy in estimating spectral quantiles based on error tolerance.

Contribution

It introduces a simple proof technique for a spectral Chernoff bound and characterizes the computational complexity of spectral estimation in local Hamiltonians.

Findings

NP-hardness for exponential error bounds

Triviality for certain polynomial error bounds

Connection to cluster expansion methods

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}.