# The complexity of LH restricted to projectors

Let's denote $$kLP_{c}$$ the promise problem where the input, given in an explicit encoding with finite number of bits, is a set $$\{p_{1},p_{2},\ldots p_{m}\}$$ of k-local projectors over a n-qbits register such that the least eigenvalue $$\varepsilon_{\min}$$ of $$\sum_{i=1}^{m}p_{i}$$ is either $$\varepsilon_{\min}\leq\varepsilon\tag{y}\label{eq:LHy-1}$$ where $$\varepsilon$$ is a parameter given in input, or $$\varepsilon_{\min}(n)\geq\varepsilon+n^{-c}\tag{n}\label{eq:LHn-1}$$ and the answer is yes (no) in the \ref{eq:LHy-1} (\ref{eq:LHn-1}) case.

That is, the $$k$$LH promise problem with hamiltonians that are projectors. Is this restriction known/studied? A circuit->projectors mapping of the type used to prove QMA1-hardness of QSAT should work in mapping to kLP the generic QMA problem. Or one could maybe reduce a kLH instance to an equivalent instance of kLP. However, how would the energy gap behave?