# Performance Analysis and Parameter Optimization of the Optimal Fixed-Point Quantum Search
## 6 Conclusions  
In summary, we have analyzed the performance and optimized the parameters of the OFPQS algorithm. We derived the least number of iterations (denoted by \(l_{opt}\)) of the OFPQS algorithm to ensure the success probability for a given lower bound of the fraction of target items no less than \(1-\delta^{2}\). Moreover, all extreme points as well as the minimum value of the success probability of the OFPQS algorithm were analyzed. In addition, we calculated the upper bound of expected queries of repeatedly executing the OFPQS algorithm to find a target item, and further analytically derived the optimal parameter that minimizes this upper bound. Compared with the minimum number of iterations given by [20], our optimal number of iterations \(l_{opt}\) has a significant reduction, e.g., when \(\delta=0.8\), almost a quarter of iterations can be saved. Our study can provide a guideline for the research and application of the OFPQS algorithm. Conceptualization, T.B. and D.H.; Writing--original draft, T.B. ; Writing--review and editing, D.H. This research was funded by the NATIONAL NATURAL SCIENCE FOUNDATION OF CHINA grant number 61801522. The authors declare no conflict of interest. The following abbreviation is used in this manuscript:  
Optimal fixed-point quantum search  
Appendix A Proof of the Extreme Properties of \(P_{L}\left(\lambda\right)\) on \(\left(0,1\right)\)  
Based on Equation (10), we can get the derivative of \(P_{L}\left(\lambda\right)\) with respect to \(\lambda\) as  
\[\frac{\partial P_{L}}{\partial\lambda} = -\frac{\delta^{2}}{2\gamma\sqrt{1-\lambda}}\frac{\sin\left[2L \arccos\left(x\right)\right]}{\sin\left[\arccos\left(x\right)\right]}, \tag{11}\]  
where \(\gamma=T_{1/L}^{-1}\left(1/\delta\right)\) and  
\[x\equiv T_{1/L}\left(1/\delta\right)\sqrt{1-\lambda}. \tag{12}\]  
Then, for any \(\lambda\in\left(0,1\right)\) and \(\delta\in\left(0,1\right]\),  
\[\frac{\partial P_{L}}{\partial\lambda}=0\Leftrightarrow\sin\left[2L\arccos \left(x\right)\right]=0\,\text{and}\,\arccos\left(x\right)\
eq 0. \tag{13}\]  
Note that \(\frac{\partial P_{L}}{\partial x}=-2L\delta^{2}\
eq 0\) when \(\arccos\left(x\right)=0\). Furthermore, we can get  
\[x=\cos\left(\frac{k\pi}{2L}\right)\in\left(0,1\right). \tag{14}\]  
Note that, \(x\
eq 1\) due to \(\arccos\left(x\right)\
eq 0\), and \(x>0\) for \(\lambda\in\left(0,1\right)\) and \(\delta\in\left(0,1\right]\). For the case \(L>1\), from Equations (10), (12), and (14), it follows that when \(k=2j-1\), \(P_{L}\left(\lambda\right)=1\), where \(j=1,2,\cdots,l\equiv\left(L-1\right)/2\), thus, the local maximum points can be obtained as  
\[\lambda_{max,j}=1-\gamma^{2}\cos^{2}\left(\frac{2j-1}{2L}\pi\right),\,j=1,2, \cdots,l. \tag{15}\]Additionally, because \(P_{L}\left(\lambda=0\right)=0\) and \(P_{L}\left(\lambda=1\right)=1\), \(P_{L}\) has the same number of local maximum points and minimum points. Thus, when \(k=2j\), \(j=1,2,\cdots,l\), the corresponding local minimum points can be found, denoted by  
\[\lambda_{min,j}=1-\gamma^{2}\cos^{2}\left(\frac{j}{L}\pi\right),\,j=1,2,\cdots,l. \tag{10}\]  
For the case \(L=1\), there is no available \(k\) for Equation (11), namely, \(P_{L}\) has no extreme points. Appendix B Proof of the Minimum of \(P_{L_{opt}}\) on \(\left[\lambda_{0},1\right)\) of Equation (20)  
Based on Equation (13), for any \(l\geq l_{opt}\), if the local minimum point \(\lambda_{min,l}\) (defined by Equation (19)) exists and \(\lambda_{min,l}\in\left[\lambda_{0},1\right)\), then the minimum of \(P_{L}\left(\lambda\right)\) with \(L=2l+1\) can be obtained as \(1-\delta^{2}\), due to \(P_{L}\left(\lambda\right)\geq 1-\delta^{2}\) for any \(\lambda\geq\lambda_{0}\), and \(P_{L}\left(\lambda_{min,j}\right)=1-\delta^{2}\) for any \(1\leq j\leq l\). Note that, \(\lambda_{min,l}=\max\left\{\lambda_{min,j}:1\leq j\leq l\right\}\).