# Performance Analysis and Parameter Optimization of the Optimal Fixed-Point Quantum Search
## 5 Discussion
### Effects of \(l_{opt}\) and \(\delta_{opt}\)  
Different from our optimal number of iterations \(l_{opt}\) of Equation (13), to achieve a success probability no less than \(1-\delta^{2}\) for any \(\lambda\geq\lambda_{0}\) in the OFPQS algorithm [20], another number of iterations can be obtained from Equation (12), denoted by  
\[l_{min}^{\text{Yoder's}}\equiv\left\lceil\frac{\ln\left(2/\delta\right)}{2 \sqrt{\lambda_{0}}}-\frac{1}{2}\right\rceil. \tag{30}\]  
As a comparison, Figure 2 shows some examples of the success probability \(P_{L}\left(\lambda\right)\) for the different number of iterations \(l\) with \(L=2l+1\). We can see that both \(l=l_{min}^{\text{Yoder's}}\) and \(l=l_{opt}\) can achieve the goal \(P_{L}\left(\lambda\right)\geq 1-\delta^{2}\) for \(\lambda\geq\lambda_{0}\); while \(l=l_{opt}-1\) can't. Therefore, \(l_{opt}\) is just the least number of iterations required. When \(\lambda_{0}\ll 1\), \(\text{arcosh}\left(1/\sqrt{1-\lambda_{0}}\right)\approx\sqrt{\lambda_{0}}\), then, from Equations (13) and (30) it follows that  
\[l_{opt}/l_{min}^{\text{Yoder's}}\approx\frac{\text{arcosh}\left(1/\delta \right)}{\ln\left(2/\delta\right)}, \tag{31}\]  
which is shown in the inset of Figure 2.  
We can see that when \(\delta\to 1\), \(l_{opt}/l_{min}^{\text{Yoder's}}\to 0\). For example, when \(\delta=0.8\), \(l_{opt}/l_{min}^{\text{Yoder's}}\approx 75.65\%\), almost a quarter of iterations can be saved.  
For any given lower bound \(\lambda_{0}\in\left(0,4/5\right]\) of the fraction of target items, by theoretical analysis we have obtained the analytical expression of the optimal \(\delta\), i.e., \(\delta_{opt}\) defined by Equation (24). In order  
Figure 2: (Color online.) The success probability \(P_{L}\) versus the fraction of target items \(\lambda\) for different number of iterations \(l\) with \(L=2l+1\), \(\lambda\geq\lambda_{0}=0.125\) (dotted vertical line) and \(P_{L}\geq 1-\delta^{2}=0.9\) (dotted horizontal line). The red dashed-dotted, blue solid, and green dashed curves correspond to \(l=l_{min}^{\text{Yoder's}}=3\), \(l=l_{opt}=2\) and \(l=l_{opt}-1\), respectively. Inset: We plot \(l_{opt}/l_{min}^{\text{Yoder's}}\) against \(\delta\) with \(\lambda_{0}\ll 1\).  
to compare the results of \(\delta=\delta_{opt}\) and \(\delta\
eq\delta_{opt}\) with \(l=l_{opt}\left(\delta\right)\), we define \(L_{E,rel}^{Up,nor}\left(\lambda_{0},\delta\right)\) to be the normalized relative value of \(L_{E}^{Up}\left(\lambda_{0},\delta\right)\) with respect to \(L_{E}^{Up}\left(\lambda_{0},\delta_{opt}\right)\), i.e.,  
\[L_{E,rel}^{Up,nor}\left(\lambda_{0},\delta\right) = \frac{L_{E,rel}^{Up}\left(\lambda_{0},\delta\right)}{\max\left\{ L_{E,rel}^{Up}\left(\lambda_{0},\delta\right):0<\delta\leq 1\right\}}, \tag{32}\]  
where  
\[L_{E,rel}^{Up}\left(\lambda_{0},\delta\right)=\ln\ln\left(L_{E}^{Up}\left( \lambda_{0},\delta\right)\right)-\ln\ln\left(L_{E}^{Up}\left(\lambda_{0}, \delta_{opt}\right)\right). \tag{33}\]  
The dependence of \(L_{E,rel}^{Up,nor}\left(\lambda_{0},\delta\right)\) on \(\lambda_{0}\) and \(\delta\) is shown in Figure 3, where the darker the color, the smaller the value. For ease of comparison, \(\delta_{opt}\) is marked by the white dashed curves. We can see that the color in the area corresponding to \(\delta_{opt}\) is darkest, which indicates that the optimal parameter \(\delta_{opt}\) indeed enables the upper bound of expected queries to reach the minimum.