  We study quantum annealing for combinatorial optimization with Hamiltonian $H
= z H_f + H_0$ where $H_f$ is diagonal, $H_0=-|\phi \rangle \langle \phi|$ is
the equal superposition state projector and $z$ the annealing parameter. We
analytically compute the minimal spectral gap as $\mathcal{O}(1/\sqrt{N})$ with
$N$ the total number of states and its location $z_*$. We show that quantum
speed-up requires an annealing schedule which demands a precise knowledge of
$z_*$, which can be computed only if the density of states of the optimization
problem is known. However, in general the density of states is intractable to
compute, making quadratic speed-up unfeasible for any practical combinatoric
optimization problems. We conjecture that it is likely that this negative
result also applies for any other instance independent transverse Hamiltonians
such as $H_0 = -\sum_{i=1}^n \sigma_i^x$.
