Why adiabatic quantum annealing is unlikely to yield speed-up

Abstract

We study quantum annealing for combinatorial optimization with Hamiltonian $H = z H_f + H_0$where$H_f$isdiagonal,$H_0=-|\phi \rangle \langle \phi|$ 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 $z_{\ast }$. 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 $H_0=-{\sum }_{i=1}^n{\sigma }_i^x$.