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We study search by quantum walk on a finite two dimensional grid. Thealgorithm of Ambainis, Kempe, Rivosh (quant-ph/0402107) takes O(\sqrt{N log N})steps and finds a marked location with probability O(1/log N) for grid of size\sqrt{N} * \sqrt{N}. This probability is small, thus amplitude amplification isneeded to achieve \Theta(1) success probability. The amplitude amplificationadds an additional O(\sqrt{log N}) factor to the number of steps, making itO(\sqrt{N} log N).  In this paper, we show that despite a small probability to find a markedlocation, the probability to be within an O(\sqrt{N}) neighbourhood (at anO(\sqrt[4]{N}) distance) of the marked location is \Theta(1). This allows toskip amplitude amplification step and leads to an O(\sqrt{log N}) speed-up.  We describe the results of numerical experiments supporting this idea, and weprove this fact analytically.

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