## 4 Quantum Exchange Lock Parallel Scheduler
### Quantum Circuit Hierarchical Strategy Based on Qelps
Although it reduces a certain number of extra gates, its local optimization strategy does not fully consider the complexity of global optimization and inter-layer interactions. This limitation is particularly evident when dealing with complex circuits, where its performance is relatively inferior to the other two algorithms. The optimization effect of the QELPS algorithm reaches 85.59%. Its deep optimal greedy strategy, based on SWAP conflict resolution, effectively handles CNOT gate constraint conflicts. By postponing the execution of certain quantum gates, it minimizes SWAP gate insertions, reducing circuit depth and overall complexity. Moreover, QELPS balances local suboptimal and optimal solutions, demonstrating strong optimization capabilities, particularly in large-scale circuits. The FJOSA algorithm achieves the most significant optimization effect, reaching 89.38%, and demonstrates strong global optimization capabilities. By employing cross level deep optimization and a global perspective, it effectively reduces the insertion of SWAP gates. This algorithm excels in optimizing multi-layer circuits, minimizing redundant operations. Compared with the QELPS algorithm, the optimization effect of FJOSA improves by 26.29%. Although FJOSA incurs a higher computational cost, its enhanced optimization performance still makes it superior to the other two algorithms in terms of compilation overhead. From the perspective of quantum bit count, the FJOSA algorithm achieves an optimization rate exceeding 80% for benchmark circuits with 4, 5, 12, 13, 14, and 15 qubits. In contrast, the QELPS algorithm exceeds 80% only in circuits with 4 and 14 qubits. This indicates that FJOSA demonstrates greater adaptability in large-scale circuits, particularly when handling complex structures. It effectively man-ages multi-level dependencies, enhances parallel execution, and improves overall circuit performance. In comparison, the QELPS algorithm shows limitations in optimizing larger-scale circuits. As shown in Table 2, the execution time of the 2 QAN, QELPS, and FJOSA algorithms on the benchmark circuits varies significantly. The 2 QAN algorithm has the longest average runtime, while the QELPS and FJOSA algorithms reduce execution time by 56.32% and 66.47%, respectively, compared to 2 QAN. The superior optimization of the FJOSA algorithm over QELPS is attributed to its use of a heuristic approach and a cost function to globally optimize SWAP gate insertion. Particularly in large-scale circuits, the FJOSA algorithm effectively balances local and global optimization through its cost function (\(\lambda=0.63\)). In small-scale circuits (four and five qubits), the QELPS algorithm reduces runtime by 60.53% compared to 2 QAN, whereas the FJOSA algorithm achieves only a 6.05% reduction. This is because the global optimization strategy of FJOSA does not fully demonstrate its advantages in simpler circuits. In contrast, for circuits with more than 10 qubits, the FJOSA algorithm achieves a 66.49% runtime reduction, significantly outperforming the QELPS algorithm. This highlights its strong optimization capability in handling complex circuits. ### Expanding the Number of Qubits

Since previous experiments primarily focused on datasets ranging from four to 16 qubits, we enhance the comprehensiveness of this study by expanding the range of quantum bits. As shown in Figure 7, the FJOSA algorithm builds upon the QELPS algorithm by incorporating a joint global hierarchical strategy. Experimental results indicate that when the parameter \(\lambda=0.63\), the optimization of circuit overhead is particularly