Search by Quantum Walks on Two-Dimensional Grid without Amplitude Amplification

Abstract

AbstractWe study search by quantum walk on a finite two dimensional grid. The algorithm of Ambainis, Kempe, Rivosh [AKR05] uses \(O(\sqrt{N \log{N}})\) steps and finds a marked location with probability O(1 / logN) for grid of size \(\sqrt{N} \times \sqrt{N}\). This probability is small, thus [AKR05] needs amplitude amplification to get Θ(1) 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 \(O(\sqrt{N})\) neighbourhood (at \(O(\sqrt[4]{N})\) distance) of the marked location is Θ(1). This allows to skip amplitude amplification step and leads to \(O(\sqrt{\log{N}})\) speed-up.