# Performance Analysis and Parameter Optimization of the Optimal Fixed-Point Quantum Search
## Appendix C Proof of the Optimal Parameter \(\delta_{opt}\) of Equation (24)  
Based on Equations (20) and (23), for a given \(\lambda_{0}\), the upper bound of the expected number of queries \(L_{E}^{Up}\) as a function of \(\delta\) can be obtained as follows:  
\[L_{E}^{Up}\left(\delta\right)= \begin{cases}\left(2l_{opt}+1\right)\,/\,\left(1 -\delta^{2}\right)\equiv L_{E,left}^{Up}\left(\delta\right),&\text{if }\delta<\sqrt{1-\lambda_{0}},\\ \left(2l_{opt}+1\right)\,/\,\lambda_{0}\equiv L_{E,right}^{Up}\left(\delta \right),&\text{if }\delta\geq\sqrt{1-\lambda_{0}},\end{cases}  \tag{13}\]  
where \(l_{opt}\) is defined by Equation (13). We can see that \(L_{E,left}^{Up}\left(\delta\right)\) is a step function with \(\delta\), i.e.,  
\[L_{E,left}^{Up}\left(\delta\right)=\frac{2m+1}{1-\delta^{2}},\,\text{if}\, \delta_{m}\leq\delta<\delta_{m-1},m\geq 1, \tag{14}\]  
due to \(l_{opt}=m\) for \(\delta\in\left[\delta_{m},\delta_{m-1}\right)\), where \(\delta_{m}=T_{2m+1}^{-1}\left(1/\delta_{cri}\right)\), consistent with Equation (25). From Equation (14) it follows that \(L_{E,left}^{Up}\left(\delta\right)\) monotonically increases on range \(\left[\delta_{m},\delta_{m-1}\right)\), then  
\[\min\left\{L_{E,left}^{Up}\left(\delta\right):\delta_{m}\leq \delta<\delta_{m-1}\right\}\right. \tag{15}\] \[= L_{E,left}^{Up}\left(\delta_{m}\right)=\frac{2m+1}{1-\delta_{m }^{2}}.\]  
In addition, \(L_{E,right}^{Up}\left(\delta\right)=1/\lambda_{0}=L_{E,left}^{Up}\left( \delta_{0}\right)\), therefore,  
\[\min\left\{L_{E}^{Up}\left(\delta\right):0<\delta<1\right\}=\min\left\{L_{E, left}^{Up}\left(\delta_{m}\right):m\geq 0\right\}. \tag{16}\]  
Note that if \(L_{E,left}^{Up}\left(\delta_{k}\right)\) (\(k>0\)) is minimum, then \(\delta_{k}\) is optimal; while if \(L_{E,left}^{Up}\left(\delta_{0}\right)\) is minimum, then arbitrary \(\delta\in\left[\delta_{0},1\right)\) is optimal. To determine \(\min\left\{L_{E,left}^{Up}\left(\delta_{m}\right):m\geq 0\right\}\), we define  
\[L_{E,left}^{Up,\approx}\left(\delta\right)\equiv\frac{\text{arcosh}\left(1/ \delta\right)}{\left(1-\delta^{2}\right)\text{arcosh}\left(1/\sqrt{1-\lambda _{0}}\right)}. \tag{17}\]  
From Equations (13), (13), and (17), it follows that  
\[L_{E,left}^{Up}\left(\delta\right) \geq L_{E,left}^{Up,\approx}\left(\delta\right), \tag{18}\] \[L_{E,left}^{Up}\left(\delta_{m}\right) = L_{E,left}^{Up,\approx}\left(\delta_{m}\right). \tag{19}\]  
Moreover, we can get the derivative of \(L_{E,left}^{Up,\approx}\left(\delta\right)\) with respect to \(\delta\), i.e.,  
\[\frac{\partial L_{E,left}^{Up,\approx}}{\partial\delta}=\frac{-\left(1- \delta^{2}\right)^{-2}g\left(\delta\right)}{\text{arcosh}\left(1/\sqrt{1- \lambda_{0}}\right)}, \tag{20}\]  
where  
\[g\left(\delta\right)\equiv\frac{\sqrt{1-\delta^{2}}}{\delta}-2\delta\ln\left( \frac{1+\sqrt{1-\delta^{2}}}{\delta}\right). \tag{21}\]  
Solving \(g\left(\delta\right)=0\) gives rise to a local minimum point, denoted by \(\delta_{min}^{left,\approx}\), which satisfies  
\[\delta_{min}^{left,\approx}=\cosh^{-1}\left(\frac{x}{2}\right), \tag{22}\]where \(x>0\) is the unique solution of Equation (28). Then, we can obtain (\(k\geq 1\))  
\[\min\left\{L_{E,left}^{Up}\left(\delta_{m}\right):m\geq 0\right\}= \begin{cases}L_{E,left}^{Up }\left(\delta_{0}\right),&\text{if}\delta_{min}^{left,\approx}\geq\delta_{0},\\ \min\left\{L_{E,left}^{Up}\left(\delta_{k}\right),L_{E,left}^{Up}\left(\delta_ {k-1}\right)\right\},&\text{if}\delta_{k}\leq\delta_{min}^{left,\approx}<\delta _{k-1}.\end{cases}