Quantum Search on the Spatial Grid

Some classical solutions to problems achieve particular speed ups when we allow those algorithms to become quantum based. The most obvious result is the ability to search a group of n elements in sub linear time. Classically, it is required to look at all of the elements, one at a time, to ensure that the marked item is or is not present. In the quantum world, thanks to Lov Grover [4], we can do this in O( √ n) steps and queries. In particular, this subroutine has proved useful for a number of other algorithms and achieving quantum lower bounds. In this paper we explore the complexity of performing this algorithm when reduced to movement on a two dimensional spatial grid. We restrict our model to a quantum robot walking along the two dimensional grid. We begin by describing the model we adapt for out algorithm in Section II. We then proceed to a discussion of Grover’s Algorithm in Section III and follow up with the latest results about search on a spatial grid in Section IV. Finally, we give our new algorithm for search on the spatial grid with results in Section V. We describe the case when there are multiple marked items being search for and how it differs from the non spatial version in Section VI and comment on the implications this gives for other problems in Section VII. We give some concluding remarks in Section VIII and discuss what needs to be done.