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2016

A simple proof of the detectability lemma and spectral gap amplification

Anshu, Anurag, Arad, Itai, Vidick, Thomas

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The detectability lemma is a useful tool for probing the structure of gapped ground states of frustration-free Hamiltonians of lattice spin models.

The lemma provides an estimate on the error incurred by approximating the ground space projector with a product of local projectors.
We provide a new, simpler proof for the detectability lemma, which applies to an arbitrary ordering of the local projectors, and show that it is tight up to a constant factor.
As an application we show how the lemma can be combined with a strong converse by Gao to obtain local spectral gap amplification: we show that by coarse-graining a local frustration-free Hamiltonian with a spectral gap $\gamma>0$ to a length scale $O(\gamma^{-1/2})$, one gets an Hamiltonian with an $\Omega(1)$ spectral gap.

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D. Aharonov, I. Arad, and T. Vidick, The Quantum PCP Conjecture , Tech. Rep. (arXiv:1309.7495, 2013) appeared as guest column in ACM SIGACT News archive Volume 44 Issue 2, June 2013, Pages 47–79
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