The complexity of LH with constant gap Kitaev's quantum equivalent of the Cook-Levin Theorem, provides a polynomial time classical reduction from a QMA verification circuit to a sum $H$ of local hamiltonians, such that the least eigenvalue of $H$ is either $\leq a$ or $\geq b$, with $b-a$ vanishing as an inverse polynomial in the size of the original problem. If the gap vanishes faster, the problem is not known to be in QMA, and if the gap is larger the problem is easier.
I read often, for example in wikipedia, that for constant gap QMA-hardness is a conjecture (equivalent to a quantum version of the PCP theorem). However, I sometimes read the opposite: for example Gharibian's phd thesis problem 1.7, or Gosset, Nagaj definition 4 for the qsat subcase.
Effectively, I don't understand why I cannot repeat each term in $H$ a polynomial number of times, and obtain an equivalent problem with constant gap. Is this reasoning naively wrong, or maybe I am not reading the PCP statement correctly. Maybe &quot;constant gap&quot; is just a fast way to say something more specific?
PS. I posted this on the quantum computing board too, where is probably more inherent. I noted the board after posting here, let me know if I have to cancel this.
Your original problem formulation is not correct. It's missing some normalization on $H$, otherwise it could be the case that all eigenvalues of $H$ are always smaller than $a$, making the problem completely trivial, so it couldn't possibly be QMA-complete.
One natural normalization is $||H||\leq 1$, which you can enforce by saying that if $H$ is expressed as the sum of $m$ local terms, then each term should have norm at most $1/m$. Now all eigenvalues of $H$ must lie in $[-1,1]$. For this normalization, the complexity of constant-gap Local Hamiltonian is conjectured to be QMA-hard. Note that this problem is NP-hard because of the classical PCP theorem.