Adiabatic quantum computation: Enthusiast and Sceptic's perspectives

Enthusiast's perspective: We analyze the effectiveness of AQC for a small rank problem Hamiltonian $H_F$ with the arbitrary initial Hamiltonian $H_I$. We prove that for the generic $H_I$ the running time cannot be smaller than $O(\sqrt N)$, where $N$ is a dimension of the Hilbert space. We also construct an explicit $H_I$ for which the running time is indeed $O(\sqrt N)$. Our algorithm can be used to solve the unstructured search problem with the unknown number of marked items. Sceptic's perspective: We show that for a robust device, the running time for such $H_F$ cannot be much smaller than $O(N/\ln N)$.