  This is not a disproof of the quantum PCP conjecture!
  In this note we use perturbation on the commuting Hamiltonian problem on a
graph, based on results by Bravyi and Vyalyi, to provide a partial no-go
theorem for quantum PCP. Specifically, we derive an upper bound on how large
the promise gap can be for the quantum PCP still to hold, as a function of the
non-commuteness of the system. As the system becomes more and more commuting,
the maximal promise gap shrinks.
  We view these results as possibly a preliminary step towards disproving the
quantum PCP conjecture. A different way to view these results is actually as
indications that a critical point exists, beyond which quantum PCP indeed
holds; in any case, we hope that these results will lead to progress on this
important open problem.
