Entangling and disentangling in Grover's search algorithm

Abstract

Entanglement is believed to be crucial in making quantum algorithms more powerful than their classical counterparts for certain computational tasks. In Grover's search algorithm, the Grover's operator/iteration G can be decomposed into two basic operators, i.e., G=RO, where O is so called the Oracle operator and R is the Reflection operator. To probe the production/depletion of entanglement from basic operator level, we investigate the roles the Oracle and the Reflection operators play in the entanglement dynamics during Grover's search algorithm application. Using geometric measure of entanglement (GME), we show that the Oracle operator is an entangling operator which almost always produces (increases) entanglement while the Reflection operator is a disentangling operator which mainly depletes (decreases) entanglement. We explicitly demonstrate that there exists a turning point during the Grover's iteration application with the following properties. Before that turning point, the entanglement is almost always increased when the Oracle operator is applied, and the effect of the Reflection operator on the level of entanglement can be almost ignored. However, after the turning point, both the Oracle and the Reflection operators play important roles to the entanglement, more exactly, the Reflection operator significantly decreases entanglement while the Oracle operator increases entanglement. All these results are carefully demonstrated.