Operator coherence dynamics in Grover's quantum search algorithm

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 $G=H^{\otimes n}P{H}^{\otimes n}O$, 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 $H^{\otimes n}$ is coherent. Second, we prove that the amount of the operator coherence of the first $H^{\otimes n}$ and the operator coherence produced or depleted by $H^{\otimes n}$ 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 $H^{\otimes n}$ 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.

Figure 1

The coherence dynamics with the search iteration. The red dashed line is the success probability ($p_k$), the black dotted line is the coherence after $k$ iterates of $G$, the blue line is the coherence when $O,H^{\otimes n},P,H^{\otimes n}$ (corresponding to $rgb]0.9,0,0.7•,\diamond ,◃,▹$) are applied in each iteration, the light-green line is the coherence just $H^{\otimes n},H^{\otimes n}$ without $O$ and $P$. Here, the target numbers are $t=2$ and the qubit numbers are $n=16$ (top) and $n=8$ (bottom).

Figure 2

(Top) The operator coherence of $G,H_O,H_P$ (corresponding to the green circle line, red dashed line, and black line). (Bottom) The suboperator coherence of each $H_O$ (red dashed line) and $H_P$ (black line) in one Grover iteration. Here the qubit numbers are $n=16$ and the target numbers are $t=2$.

Figure 3

The coherence dynamics with iteration number $k$ for $n=16,t=4$. (Top) The target state $|{\chi }_1\rangle$ is a product state; (bottom) $|{\chi }_1\rangle$ is an entangled state. The red dashed line is the success probability ($p_k$), the blue line is the coherence when $O,H^{\otimes n},P,H^{\otimes n}$ are applied in each iteration, the light-green line is the coherence just for applying $H^{\otimes n},H^{\otimes n}$ without $O$ and $P$.

Figure 4

The operator coherence of $G,H_O,H_P$ with iteration number $k$ for $n=16,t=4$. (Top) The target state $|{\chi }_1\rangle$ is a product state; (bottom) $|{\chi }_1\rangle$ is an entangled state. The green circle line, the red dashed line, and the black line correspond to the variations of the operator coherence of $G,H_O,H_P$, respectively.

Figure 5

The suboperator coherence of each $H_O$ and $H_P$ in one Grover iteration for $n=16,t=4$. (Top) The target state $|{\chi }_1\rangle$ is a product state; (bottom) $|{\chi }_1\rangle$ is an entangled state. The red dashed line and the black line correspond to the variations of suboperator coherence of each $H_O$ and $H_P$, respectively.