Dynamics of quantum adiabatic computation

We present an analysis of the performance of quantum adiabatic evolution algorithms (QAAs) for random combinatorial optimization problems (COPs) using a simple model of a COP that is directly related to the NP-complete Number Partitioning Problem. A key element of our approach is the problem landscape correlation; that is, the conditional distribution of the set of strings obtained from a given string by a fixed number of bit flips. We use this entity to describe QAA as a quantum local search and show that in our example of COP, the dynamics of the algorithm can be described as a 1D quantum diffusion process. The ground state remains extended in the energy space, thus leading to an exponential complexity for the algorithm. We discuss the possible generalizations of the presented analysis and the limitations of QAA.