# Performance Analysis and Parameter Optimization of the Optimal Fixed-Point Quantum Search
## 3 Performance Analysis of the OFPQS Algorithm  
After deep theoretical analysis, we can obtain three new properties on the number of iterations and success probability of the OFPQS algorithm, which are as follows.  
**Property 1**.: _Given a lower bound \(\lambda_{0}\) of the fraction \(\lambda\) of target items, the least exact number of iterations \(l\) (denoted by \(l_{opt}\)) of the OFPQS algorithm that enables the success probability \(P_{L}\left(\lambda\right)\geq 1-\delta^{2}\) for any \(\lambda\geq\lambda_{0}\) with \(L=2l+1\), can be given in the form_  
\[l_{opt}=\left\lceil\frac{\text{arcosh}\left(1/\delta\right)}{2\text{arcosh} \left(1/\sqrt{1-\lambda_{0}}\right)}-\frac{1}{2}\right\rceil. \tag{13}\]  
**Proof.** Based on Equation (10), for arbitrary \(\lambda\in\left(0,1\right)\) and \(\delta\in\left(0,1\right]\), a necessary and sufficient condition to make the success probability \(P_{L}\left(\lambda\right)\geq 1-\delta^{2}\) can be given as follows,  
\[\left|T_{1/L}\left(1/\delta\right)\right|\sqrt{1-\lambda}\leq 1, \tag{14}\]  
due to the fact that  
\[\left|T_{L}\left(x\right)\right|\leq 1\Leftrightarrow\left|x\right|\leq 1. \tag{15}\]  
Note that if \(\left|x\right|>1\), then, \(\text{arcosh}\left(\left|x\right|\right)>0\) and \(\left|T_{L}\left(x\right)\right|=\cosh\left(L\text{arcosh}\left(\left|x\right| \right)\right)>1\); if \(\left|x\right|\leq 1\), then, \(\left|T_{L}\left(x\right)\right|=\left|\cos\left(L\arccos\left(x\right) \right)\right|\leq 1\). Due to \(T_{1/L}\left(1/\delta\right)\geq 1\) and \(\cosh\left(x\right)\) monotonically increases for \(x\geq 0\), Equation (14) can be further reduced to  
\[L \geq \frac{\text{arcosh}\left(1/\delta\right)}{\text{arcosh}\left(1/ \sqrt{1-\lambda}\right)}. \tag{16}\]Then, according to \(L=2l+1\), we have  
\[l\geq\left\lceil\frac{\text{arcosh}\left(1/\delta\right)}{2\text{arcosh}\left(1/ \sqrt{1-\lambda}\right)}-\frac{1}{2}\right\rceil\,. \tag{17}\]  
Therefore, to ensure that \(P_{L}\left(\lambda\right)\geq 1-\delta^{2}\) for any \(\lambda\geq\lambda_{0}\), the least number of iterations, expressed by Equation (13), can be finally obtained.  
**Property 2**.: _The extreme properties of the success probability \(P_{L}\left(\lambda\right)\) as a function of \(\lambda\in\left(0,1\right)\) can be given as follows (see Proof in Appendix A):_  
_For \(\delta\in\left(0,1\right]\), when \(L>1\), \(P_{L}\left(\lambda\right)\) has \(\left(L-1\right)/2\equiv l\) local maximum points_  
\[\lambda_{max,j}=1-\gamma^{2}\cos^{2}\left(\frac{2j-1}{2L}\pi\right)\,,\,j=1,2,\cdots,l, \tag{18}\]  
_and \(l\) local minimum points_  
\[\lambda_{min,j}=1-\gamma^{2}\cos^{2}\left(\frac{j}{L}\pi\right)\,,\,j=1,2, \cdots,l. \tag{19}\]  
_When \(L=1\), there are no local extreme points._  
**Property 3**.: _The success probability \(P_{L}\left(\lambda\right)\) with \(L=2l_{opt}+1\equiv L_{opt}\) has a minimum value (denoted by \(P_{min}\)) on the range \(\lambda\in\left[\lambda_{0},1\right)\) for \(\lambda_{0}\leq 4/5\), which can be written as (for a proof, see Appendix B)_  
\[P_{min} \equiv \min\left\{P_{L_{opt}}\left(\lambda\right):\lambda\geq\lambda_{ 0}\right\} \tag{20}\] \[= \left\{\begin{array}{ll}1-\delta^{2},&\text{if }\delta<\sqrt{1- \lambda_{0}},\\ \lambda_{0},&\text{if }\delta\geq\sqrt{1-\lambda_{0}}.\end{array}\right.\]  
_Note that the case \(\lambda_{0}>4/5\) can already be well disposed of by classical search._