Finding structural anomalies in star graphs using quantum walks: a general approach

In previous papers about searches on star graphs several patterns have been made apparent; the speed up only occurs when graphs are 'tuned' so that their time step operators have degenerate eigenvalues, and only certain initial states are effective. More than that, the searches are never faster than O(root N) time. In this paper the problem is defined rigorously, the causes for all of these patterns are identified, sufficient and necessary conditions for quadratic-speed searches for any connected subgraph are demonstrated, the tolerance of these conditions is investigated, and it is shown that ( unfortunately) we can do no better than O(root N) time. Along the way, a useful formalism is established that may be useful in future work involving highly symmetric graphs.