  We study search by quantum walk on a finite two dimensional grid. The
algorithm 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 is
needed to achieve \Theta(1) success probability. The amplitude amplification
adds an additional O(\sqrt{log N}) factor to the number of steps, making it
O(\sqrt{N} log N).
  In this paper, we show that despite a small probability to find a marked
location, the probability to be within an O(\sqrt{N}) neighbourhood (at an
O(\sqrt[4]{N}) distance) of the marked location is \Theta(1). This allows to
skip amplitude amplification step and leads to an O(\sqrt{log N}) speed-up.
  We describe the results of numerical experiments supporting this idea, and we
prove this fact analytically.
