# Short proof of a spectral Chernoff bound for local Hamiltonians

@article{Abrahamsen2020ShortPO, title={Short proof of a spectral Chernoff bound for local Hamiltonians}, author={Nilin Abrahamsen}, journal={arXiv: Quantum Physics}, year={2020} }

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…

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