Why adiabatic quantum annealing is unlikely to yield speed-up

Abstract

We study quantum annealing for combinatorial optimization with Hamiltonian $H = H_0 + z H_f$ where $H_f$ is diagonal, $H_0=-\ket{\phi}\bra{\phi}$ is the equal superposition state projector and $z$ the annealing parameter.
We analytically compute the minimal spectral gap, which is $\Omega(1/\sqrt{N})$ 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 combinatorial optimization problems. 
We conjecture that it is likely that this negative result also applies for any other instance independent transverse Hamiltonians such as $H_0 = -\sum_{i=1}^n \sigma_i^x$.