## 4 Quantum Exchange Lock Parallel Scheduler
### Deep Optimal Greedy Algorithm Layering Strategy Based on Swap Conflict  
**Theorem 1**. : _SWAP Conflict. Consider a quantum circuit \(Q=\{q_{0},q_{1},\ldots,q_{n}\}\) that includes two CNOT gates, \(\text{CNOT}q_{i},q_{e}\) and \(\text{CNOT}q_{u},q_{v}\). If \(\text{CNOT}q_{i},q_{e}\) fails to meet the topological constraint, a SWAP operation, \(\text{SWAP}q_{i},q_{v}\), is introduced to alter the execution order of the qubits. However, this SWAP operation cannot operate on the same qubit \(q_{i},q_{v}\) simultaneously with both CNOT gates._  
**Theorem 2**. : _Local suboptimal solution. Consider a quantum circuit \(Q=\{q_{0},q_{1},\ldots,q_{n}\}\) and a circuit layer \(L=\{l_{0},l_{1},\ldots,l_{m}\}\) that contain quantum gates \(\text{CNOT}q_{a},q_{b}\) and \(\text{CNOT}q_{c},q_{d}\). Additionally, a quantum gate \(\text{CNOT}q_{u},q_{v}\) is present in layer \(l_{e}\). When \(\text{CNOT}q_{a},q_{b}\) fails to meet the topological constraint, the \(\text{SWAP}q_{a},q_{c}\) operation is introduced to alter the execution order of the qubits. However, \(\text{SWAP}q_{a},q_{c}\) cannot operate on the same qubit involved in both \(\text{CNOT}q_{a},q_{b}\) and \(\text{CNOT}q_{c},q_{d}\) within the layer. Consequently, \(\text{CNOT}q_{c},q_{d}\) is temporarily postponed to layer \(l_{e}\)._  
The first layering process employs a deep optimal greedy algorithm based on SWAP conflicts for quantum circuit partitioning. As shown in Equation (2), \(\text{SWAP}_{total}\) represents the total inserted SWAP gates, where \(\text{SWAP}_{i}\) corresponds to the SWAP count in the \(i^{th}\) layer, ensuring \(\text{SWAP}_{total}\) equals the summation of \(\text{SWAP}_{i}\) across all layers. Traditional greedy algorithms are observed to resolve mapping conflicts immediately at each layering stage, leading to frequent \(\text{SWAP}_{i}\) insertions and increased circuit complexity. In contrast, the execution of specific quantum gates is intentionally postponed when conflicts are encountered in our approach, enabling temporary acceptance of local suboptimal configurations. While immediate conflict resolution (a locally optimal strategy) substantially elevates \(\text{SWAP}_{total}\) in multilayered circuits, the proposed method strategically delays partial operations to prioritize global optimization. Conflicts in specific layers are purposefully maintained through controlled \(\text{SWAP}_{i}\) allocations to achieve this balance. The algorithm dynamically selects resolution strategies between optimal and suboptimal choices based on layer-specific parameters, thereby enhancing operational adaptability. This systematic compromise between localized SWAP gate management and circuit-wide efficiency ensures robust quantum circuit performance without logical integrity loss. \[\text{SWAP}_{total}=\sum_{i=1}^{n}\text{SWAP}_{i} \tag{2}\]  
The specific operation steps are as follows:  
**Step 1:** Quantum gates violating operational constraints are identified. The SWAP gate insertion schemes generated from these constrained gates are analyzed, forming the foundational basis for subsequent stratification. **Step 2:** A local suboptimal solution is selected rather than enforcing immediate SWAP gate insertion, allowing strategic postponement of conflicting quantum operations. **Step 3:** Residual quantum gates are progressively incorporated during layering iterations, with dual evaluation of both locally optimal and suboptimal configurations during insertion decisions. **Step 4:** Non-conflicting quantum gates relative to existing components in the current layer are directly integrated; those exhibiting conflicts are systematically postponed for reprocessing. **Step 5:** For quantum gates violating CNOT adjacency constraints, their SWAP insertion schemes are combined with prior configurations to establish optimized stratification baselines.