# Optimization Strategies in Quantum Machine Learning: A Performance Analysis
## 5 Comparative Analysis
### ADAM Optimizer  
We utilized the ADAM optimizer, a gradient-based algorithm with parameter-shift gradients. The default learning rate of 0.001 was applied, and the optimization process was conducted for a maximum of 22 iterations. The confusion matrix resulting from the ADAM optimizer is presented below:  
\[ \begin{bmatrix}1&28\\ 1&31\end{bmatrix}  \tag{8}\]  
True positive (TP): Only one instance was correctly predicted as positive. False positive (FP): Twenty-eight instances were incorrectly predicted as positive. False negative (FN): One instance was incorrectly predicted as negative. True negative (TN): Thirty-one instances were correctly predicted as negative. Using the confusion matrix values, the performance metrics for the ADAM optimizer were calculated as follows:  
\[\text{Accuracy}=\frac{31+1}{31+28+1+1}=0.524=52\%\]  
\[\text{Precision}=\frac{31}{31+28}=0.525=53\%\]  
\[\text{Recall}\ (\text{Sensitivity})=\frac{31}{31+1}=0.9688=97\%\]  
\[\text{F1 score}=2\times\frac{0.5254\times 0.9688}{0.5254+0.9688}=0.6808=68\%\]  
The model trained using the ADAM optimizer required 10 min, the longest training time among the three optimizers. The performance metrics for all three optimizers are summarized below. Table 1 presents a comparative analysis of three optimizers across various performance metrics, revealing significant differences in their effectiveness. The confusion matrices are compared below to provide a detailed evaluation of each model's classification performance, highlighting how well each optimizer distinguishes between classes:  
An interpretation of the results in Table 2 shows the following:  
COBYLA exhibits the best performance with the highest Correctly Identified Positive (25) and Correctly Identified Negative (31) counts, and the lowest Incorrectly Identified Negative (1) and Incorrectly Identified Positive (4) counts. This suggests that it is the most accurate optimizer for this specific problem. L-BFGS-B performs slightly worse than COBYLA, with a slightly lower Correctly Identified Positive (24) and Correctly Identified Negative (30) count and slightly higher Incorrectly Identified Positive (5) and Incorrectly Identified Negative (2) counts. However, it still performs considerably better than ADAM. \begin{table}
\begin{tabular}{c c c c c c c c} \hline
**Optimizer** & **Accuracy** & **Precision** & **Recall** & **F1 Score** & **Training Time** & **Converges** & **Loss** \\ \hline COBYLA & 92\% & 89\% & 97\% & 93\% & 1 & 100\% & 8.20\% \\ L-BFGS-B & 89\% & 86\% & 94\% & 90\% & 6 & 100\% & 11.84\% \\ ADAM & 52\% & 53\% & 97\% & 68\% & 10 & 100\% & 47.54\% \\ \hline \end{tabular}
\end{table} 
Table 1: Performance metrics of optimizers. These findings highlight the critical role played by optimizer selection in achieving both high performance and computational efficiency in quantum machine learning models. Optimizer Performance Analysis  
A comparative evaluation of the three optimizers--COBYLA, L-BFGS-B, and ADAM--reveals distinct strengths and weaknesses in their performance across various metrics. This analysis focuses on accuracy, precision, recall, and F1 score, providing a comprehensive understanding of the optimizers' suitability for the quantum classification model applied to the Cleveland dataset. 1. Accuracy: COBYLA emerged as the top performer with an impressive accuracy rate of 92%, making it the most reliable optimizer for this task. L-BFGS-B followed with a respectable accuracy of 89%, while ADAM lagged significantly behind at 52%. This stark difference underscores COBYLA's superiority in correctly classifying instances. 2. Precision: Precision measures the proportion of true positive predictions among all positive classifications. COBYLA led once again with a precision rate of 89%, demonstrating its ability to effectively minimize false positives. L-BFGS-B achieved a slightly lower precision of 86%, maintaining reasonable performance. In contrast, ADAM exhibited a markedly low precision of 53%, indicating a high rate of incorrect positive predictions. 3.