量子近似最佳化演算法[1]已被提出來解決組合問題，例如最大割問題，但它無法直接應於有約束條件的問題。先前的研究使用懲罰函式法或量子交替算子擬設[2]來處理這些約束條件。在這篇論文中，我們以最小頂點覆蓋問題作為測試案例，分析了將約束條件引入量子近似最佳化演算法的兩種方法之間的差異。同時，我們遵循Brayvi等人[3]的方法，通過遞迴方法增強量子交替算子擬設並使用古典電腦上的模擬器來測試算法的性能。我們發現，使用遞迴方法顯著提高了性能，優於古典貪婪算法。最後，我們將深度為1的量子算法轉化為遞迴式古典近似最佳化算法，並在更大的圖上進行了測試。結果顯示，在邊數較多的隨機圖上，遞迴式古典近似最佳化算法的性能在我們的測試條件下優於貪婪算法和遺傳算法。

Quantum Approximation Optimization Algorithm (QAOA) [1] has been proposed to solve combinatorial problems, such as maximum cut problem, but it cannot be directly applied to problems with constraints. Previous research has used the penalty method or Quantum Alternating Operator Ansatz (QAOAnsatz) [2] to handle the constraints. In this work, Using the Minimum Vertex Cover (MVC) problem as an test case, we analyzed the differences between two methods of incorporating constraints into QAOA. Also, we followed the approach of Brayvi et al. [3], enhancing QAOAnsatz with a recursive method and using a classical simulator to test the algorithm’s performance. We found that using the recursive method significantly improves performance, outperforming the classical greedy algorithm. Finally, we transformed our quantum algorithm at depth-1 into a Recursive Classical Approximation Optimization Algorithm (RCAOA) and tested it on larger graphs. The results on larger graphs show that when the graph has many edges, RCAOA’s performance on random graphs is better than that of the greedy algorithm and genetic algorithm under our test conditions.

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