Operator coherence dynamics in Grover's quantum search algorithm

Abstract

Coherence is one of the most basic concepts and resources in quantum information. To clear the role coherence plays on the essential operator level in Grover's search algorithm, here we discuss the coherence dynamics of the state after each basic operator is applyied. As it is known, Grover's search algorithm repeats the application of Grover operator $G$, which can be decomposed into $G={H}^{\ensuremath{\bigotimes}n}P{H}^{\ensuremath{\bigotimes}n}O$, where $H$ is Hadamard operator, $P$ is the condition phase-shift operator, and $O$ is the oracle operator. First, we show that $O$ and $P$ are incoherent operators while ${H}^{\ensuremath{\bigotimes}n}$ is coherent. Second, we prove that the amount of the operator coherence of the first ${H}^{\ensuremath{\bigotimes}n}$ and the operator coherence produced or depleted by ${H}^{\ensuremath{\bigotimes}n}$ depends not only on the size of the database and the success probability, but also on target states. Moreover, the amount of operator coherence is larger when the superposition state of targets is entangled rather than product. Third, we show that the two ${H}^{\ensuremath{\bigotimes}n}$ have different effects on coherence that one produces coherence and the other depletes coherence, and the depletion plays a major role. Therefore, the coherence is vibrating during the search process and the overall effect is that coherence is in depletion.