Search by quantum walks on two-dimensional grid without amplitude amplification

Abstract

We study search by quantum walk on a finite two dimensional grid. The algorithm of Ambainis, Kempe, Rivosh [AKR05] uses O( √ N logN) steps and finds a marked location with probability O(1/ logN) for grid of size √ N × √ N. This probability is small, thus [AKR05] needs amplitude amplification to get θ(1) probability. The amplitude amplification adds an additional O( √ logN) factor to the number of steps, making it O( √ N logN). In this paper, we show that despite a small probability to find a marked location, the probability to be within O( √ N) neighbourhood (at O( 4 √ N) distance) of the marked location is θ(1). This allows to skip amplitude amplification step and leads to O( √ logN) speed-up. © Springer-Verlag Berlin Heidelberg 2013.