Grover search under localized dephasing

TL;DR

This paper investigates how localized dephasing noise affects the efficiency of Grover's quantum search algorithm, revealing conditions under which the quantum advantage is preserved or lost, and connecting results to quantum walks and graph searches.

Contribution

It introduces a model of localized partial dephasing noise in Grover search and analyzes the conditions for maintaining quantum speedup under such noise.

Findings

Quadratic speedup is retained when noise affects a small subspace and the target is unaffected.

Speedup is lost if the noise affects the target or the affected subspace is large.

When the target is unaffected, the noise rate must scale as 1/√N to preserve speedup.

Abstract

Decoherence in quantum searches, and in the Grover search in particular, has already been extensively studied, leading very quickly to the loss of the quadratic speedup over the classical case, when searching for some target (marked) element within a set of size $N$. The noise models used were, however, global. In this paper we study Grover search under the influence of localized partially dephasing noise of rate $p$. We find, that in the case when the size $k$ of the affected subspace is much smaller than $N$, and the target is unaffected by the noise, namely when $kp\ll\sqrt{N}$, the quadratic speedup is retained. Once these restrictions are not met, the quadratic speedup is lost. In particular, if the target is affected by the noise, the noise rate needs to scale as $1/\sqrt{N}$ in order to keep the speedup. We observe also an intermediate region, where if $k\sim N^\mu$ and the target is unaffected, the speedup seems to obey $N^\mu$, which for $\mu>0.5$ is worse than the quantum, but better than the classical case. We put obtained results for quantum searches also into perspective of quantum walks and searches on graphs.