On degradation of Grover's search under collective phase flips in queries to the oracle

TL;DR

This paper investigates how collective phase flip noise affects Grover's search algorithm, revealing that even small errors significantly reduce success probability and exploring the relationship between quantum coherence and algorithm performance.

Contribution

It introduces a simple model of collective phase distortions in Grover's algorithm and analyzes their impact on success probability and quantum coherence.

Findings

Success probability decreases significantly with phase errors.

Success probability quickly approaches 50%, indicating a mixed state.

Trade-off relations between quantum coherence and success probability are identified.

Abstract

We address the case, when querying to the oracle in Grover's algorithm is exposed to noise including phase distortions. The oracle-box wires can be altered by an opposite party that tries to prevent correct receiving data from the oracle. This situation reflects an experienced truth that any access to prophetic knowledge cannot be common and direct. To study the problem, we introduce a simple model of collective phase distortions on the base of phase damping channel. In the model used, the success probability is not altered via the oracle-box wires {\it per se}. Phase distortions of the considered type can hardly be detected via any one-time query to the oracle. However, the success probability is significantly changed, when such errors are introduced as an intermediate step into the Grover iteration. We investigate the success probability with respect to variations the parameter that characterizes the amount of phase errors. It turns out that the success probability is decreased significantly even if the error amount is not very high. Moreover, this probability quickly reduces to the value of one half, which corresponds to the completely mixed state. We also study trade-off relations between quantum coherence and the success probability in the presence of noise of the considered type.