Quantum searching's underlying SU(2) structure and its quantum decoherence effects

TL;DR

This paper reveals that quantum search operations are governed by SU(2) group structure, allowing precise bounds on steps and detailed analysis of decoherence effects, which can impair or invalidate the search process.

Contribution

It uncovers the SU(2) group structure underlying quantum search algorithms and analyzes how decoherence impacts their effectiveness.

Findings

Quantum search operations form an SU(2) group element.

Exact bounds on the number of steps are derived.

Decoherence can either invalidate or worsen the search probability.

Abstract

The search operation for a marked state by means of Grover's quantum searching algorithm is shown to be an element of group SU(2) which acts on a 2-dimensional space spanned by the marked state and the unmarked collective state. Based on this underlying structure, those exact bounds of the steps in various quantum search algorithms are obtained in a quite concise way. This reformulation of the quantum searching algorithm also enables a detailed analysis of the decoherence effects caused by its coupling with an environment. It turns out that the environment will not only make the quantum search invalid in case of complete decoherence, where the probability of finding the marked state is unchanged, but also it may make the quantum search algorithm worse than expected: It will decrease this probability when the environment shows its quantum feature.