Purpose: This paper focuses on enhancing the existing Cuckoo Search Algorithm (CSA) by integrating it with the Quantum Computing (QC) mechanism to develop the QC-CSA hybrid algorithm for metaheuristic optimizations in the area of the Traveling Salesman Problem (TSP). Further, this research work provides a second level of hybridization by combining the first-level hybrid QC-CSA algorithm with five Classification Algorithms (CA), including Decision Tree (DT), K-Nearest Neighbors (KNN), Logistic Regression (LR), Random Forest (RF), and Support Vector Machine (SVM), to design and develop algorithms: QC-CS-DT, QC-CS-KNN, QC-CS-LR, QC-CS-RF, and QC-CS-SVM, respectively, under the QC-CS-CA category. Dataset: More than fifty cities were considered for the TSP with features including distance and deadlines. Performance Parameters: Averages of Best Fitness (Avg. BF), Execution Time (Avg. ET), and Memory Usage (Avg. MU) were used to compare the existing CSA with QC-CSA and all the algorithms under the QC-CS-CA category. Experiment: A range of ten nests to one hundred nests was considered to embed the CSA essence and complete all the cities at least once, as per the requirement of TSP. Results: The proposed algorithms under the QC-CS-CA category, including QC-CS-DT, QC-CS-SVM, and QC-CS-RF, provide best results with improvement of 44.43%, 7.14% and 0.36% considering Avg. BF, Avg. ET and Avg. MU, compared to the existing CSA. Conclusion: The integration of QC, CSA, and CA algorithms provided optimized results for the TSP. Future Work: To ensure further optimization, the proposed algorithms will be embedded with neural-networks as a part of future work.