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

TL;DR

This paper demonstrates that quantum walks on a two-dimensional grid can find a marked location more efficiently by leveraging the probability of being near the target, eliminating the need for amplitude amplification and achieving a speed-up.

Contribution

It shows that amplitude amplification can be skipped in quantum walk search on 2D grids by analyzing near-miss probabilities, leading to faster algorithms.

Findings

Probability of being near the marked location is constant.

Skipping amplitude amplification reduces complexity from O(√N log N) to O(√N)

Numerical and analytical results support the near-miss probability advantage.

Abstract

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.