## 4 Quantum Exchange Lock Parallel Scheduler
### Quantum Circuit Hierarchical Strategy Based on Qelps
Although optimizing more layers simultaneously can further reduce SWAP gate insertions, it also significantly increases computational complexity. Therefore, by employing \(Q(i)^{l}\), a restrictive effect is imposed. By reasonably selecting the number of optimized layers, a balance can be achieved between reducing the insertion of SWAP gates and controlling algorithm complexity. This approach ensures efficient circuit execution while minimizing computational overhead. One critical aspect of the FJOSA algorithm is its computational scalability under increasing circuit depth and qubit count, which was not explicitly discussed in earlier sections. As the number of circuit layers \(L\) increases, and more quantum bits \(n\) are involved in optimization, the number of possible SWAP insertion combinations across multiple layers grows rapidly. Specifically, the search space complexity increases approximately as \(O\big{(}n^{2}.L^{2}\big{)}\), since each additional layer allows for more interaction permutations among qubit pairs. This effect becomes even more pronounced when gate congestion or hardware constraints introduce dependency chains across layers, which further increase the cost of evaluating and selecting valid gate placements. In FJOSA, the multi-layer scheduling process evaluates several candidate configurations to balance gate satisfaction and SWAP reduction. The algorithm's complexity scales quadratically with both the number of active qubits and optimization layers. While this improves global performance, it also introduces overhead as circuit depth increases. Therefore, the number of layers per optimization round must be bounded in practice, particularly for circuits with over 20 qubits or 50 layers. This highlights the need for scalable heuristics when extending FJOSA to larger circuits. The specific operations of the FJOSA algorithm are as follows:

1. **Initial Stratification:** The QELPS algorithm is used to stratify quantum circuits. 2. **Screening the Optimal SWAP Combination:** In the first layer, all SWAP gate combination solutions that solve the CNOT gate constraint are identified and compared. The solution with the least number of SWAP gates is selected. 3. **Adjacent Layer Optimization:** In the second layer, the best SWAP gate solution from the previous step is compared with the solutions in subsequent layers. The solution with the fewest SWAP gates is chosen. 4. **Functional Judgment and Joint Optimization:** Steps (2) and (3) are applied to the first \((e+l)\) layers, where \((e=0,\ 1,\ 2,\ldots,m;l=0,\ 1,\ 2,\ldots,n)\). If the cost function \((Cost(i,l,e))\) for the \(e+l\) layer is greater than zero, the first and \(e+l-1^{th}\) layers are optimized. In this case, the optimization starts from the initial layer, specifically when \(e=0\). 5. **Overall Layer Optimization:** Once the mapping of the previous layer is satisfied, the optimization continues for all subsequent layers until the SWAP gate optimization for all layers is completed. The pseudo code of the FJOSA algorithm is as follows (Algorithm 2). Input:

_Layer_i_ : The \(i^{th}\) layer of quantum circuit operation data

_Swap_i_j_ : The number of SWAP gates required for the \(j^{th}\) scheme in the \(i^{th}\)layer.