Quantum Walks on Two-Dimensional Grids with Multiple Marked Locations

The running time of a quantum walk search algorithm depends on both the structure of the search space graph and the configuration of marked locations. While the first dependence has been studied in a number of papers, the second dependence remains mostly unstudied. We study search by quantum walks on the two-dimensional grid using the algorithm of Ambainis, Kempe and Rivosh [AKR05]. The original paper analyses one and two marked locations only. We move beyond two marked locations and study the behaviour of the algorithm for an arbitrary configuration of marked locations. In this paper, we prove two results showing the importance of how the marked locations are arranged. First, we present two placements of k marked locations for which the number of steps of the algorithm differs by a factor of $$\varOmega \sqrt{k}$$. Second, we present two configurations of k and $$\sqrt{k}$$ marked locations having the same number of steps and probability to find a marked location.