Why adiabatic quantum annealing is unlikely to yield speed-up

We study quantum annealing for combinatorial optimization with Hamiltonian H = H 0 + z H f where H f is diagonal, H 0 = − | ϕ ⟩ ⟨ ϕ | is the equal superposition state projector and z the annealing parameter. We analytically compute the minimal spectral gap, which is O 1 / 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 = − ∑ i = 1 n σ i x .