Adiabatic quantum computation: Enthusiast and Sceptic's perspectives

Enthusiast’s perspective: We analyze the effectiveness of AQC for a small rank problem Hamiltonian HF with the arbitrary initial Hamiltonian HI . We prove that for the generic HI the running time cannot be smaller than O( √ N), where N is a dimension of the Hilbert space. We also construct an explicit HI for which the running time is indeed O( √ 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 HF cannot be much smaller than O(N/ lnN).