Coherence depletion in the Grover quantum search algorithm

We investigate the role of quantum coherence depletion (QCD) in the Grover search algorithm (GA) by using several typical measures of quantum coherence and quantum correlations. By using the relative entropy of coherence measure (${\mathcal{C}}_r$), we show that the success probability depends on the QCD. The same phenomenon is also found by using the $l_1$ norm of coherence measure (${\mathcal{C}}_{l_1}$). In the limit case, the cost performance is defined to characterize the behavior about QCD in enhancing the success probability of GA, which is only related to the number of searcher items and the scale of the database, regardless of using ${\mathcal{C}}_r$ or ${\mathcal{C}}_{l_1}$. In the generalized Grover search algorithm (GGA), the QCD for a class of states increases with the required optimal measurement time. In comparison, the quantification of other quantum correlations in GA, such as pairwise entanglement, multipartite entanglement, pairwise discord, and genuine multipartite discord, cannot be directly related to the success probability or the optimal measurement time. Additionally, we do not detect pairwise nonlocality or genuine tripartite nonlocality in GA since Clauser-Horne-Shimony-Holt inequality and Svetlichny's inequality are not violated.

Figure 1

An illustration to show the first two steps of GA. First, initialize the $n$-qubit database to $|{\psi }_0\rangle$; second, $O$ reflects $|{\psi }_0\rangle$ according to $|X^{\perp }\rangle$ and $D$ reflects $O|{\psi }_0\rangle$ according to $|\psi \rangle$. Consequently, one whole iteration $G$ turns the vector before iteration anticlockwise by an angle $\alpha$.

Figure 2

Evolution of coherence in GA for the whole 11-qubit system with $j$ (from 1 to 10) solutions.

Figure 3

Coherence depletion $\mathrm{\Delta }{\mathcal{C}}_r^{\mathrm{GGA}}$ vs optimal measurement time $r_{\mathrm{opt}}^{\mathrm{GGA}}$ for the initial states of $|{\varphi }_0\rangle$ in GGA.

Figure 4

The evolutions for the entanglement in the case of 11-qubit system. The pairwise entanglement is depicted by orange diamonds while the entanglement of the whole 11-qubit system is shown by pink points. The blue squares represent the success probability.

Figure 5

The evolutions for the discord in the case of 11-qubit system. The pairwise discord is depicted by green diamonds while the genuine quantum correlation of the whole 11-qubit system is shown by red points. The blue squares represent the success probability.