Why adiabatic quantum annealing is unlikely to yield speed-up

We study quantum annealing for combinatorial optimization with Hamiltonian $H=z{H}_f+H_0$ where $H_f$ is diagonal, $H_0=-\mathrm{\mid }\phi ⟩⟨\phi \mathrm{\mid }$ is the equal superposition state projector and $z$ the annealing parameter. We analytically compute the minimal spectral gap as $\mathcal{O}(1\mathrm{/}\sqrt{N})$ with $N$ the total number of states and its location $z_{\ast }$. We show that quantum speed-up requires an annealing schedule which demands a precise knowledge of $z_{\ast }$, 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 $H_0=-{\sum }_{i=1}^n{\sigma }_i^x$.