遞迴量子近似最佳化演算法（RQAOA）是一種非局部的量子近似最佳化演算法（QAOA）的變體，具有與經典演算法競爭的潛力。基於前景看好的RQAOA，我們開發了兩種RQAOA的變體，希望能獲得更好的性能或防止RQAOA陷入一些不夠好的解答。第一種變體是受到RQAOA$_1$啟發的經典演算法（RICA），其針對QAOA$_1$的原始公式中的不想要的部分進行了修改。我們將RICA與原始的RQAOA$_1$在隨機的實例及對於RQAOA$_1$較難的實例上進行了比較，結果顯示平均近似比有所改善。第二種變體是隨機化RQAOA（RRQAOA），其在RQAOA選擇頂點對的策略中增加了更多隨機性。我們展現了RRQAOA可以防止陷入一些不夠好的解答，並展示了其在無權重的60節點圖和有權重的80節點圖上與最先進的Goemans-Williamson演算法競爭的能力。

Recursive quantum approximate optimization algorithm (RQAOA) is a non-local variant of quantum approximate optimization algorithm (QAOA) that has the potential to compete with classical algorithm. Based on the promising RQAOA, we developed two variants of RQAOA in hope of getting even better performance or prevent RQAOA from being stuck on some not good enough solutions. The first variant is the RQAOA$_1$ inspired classical algorithm (RICA) which tailored the unwanted part of the original formula of QAOA$_1$. We compared RICA with the original RQAOA$_1$ on random instances and hard instances of RQAOA$_1$ and showed an improvement on average approximation ratio. The second variant is the randomized RQAOA (RRQAOA) which adds more randomness into the strategy of selecting vertices pair of RQAOA. We showed that RRQAOA can prevent getting stuck and showed it's ability to compete with the state-of-the-art Goemans-Williamson algorithm on unweighted 60 nodes graphs and weighted 80 nodes graphs.

Abstract (Chinese) I
Abstract II
Contents III
List of Figures IV
1 Introduction 1
2 Background 4
2.1 weighted Maximum Cut (Max-Cut) problem . . . . . . . . . . . . . 4
2.2 Quantum Approximate Optimization Algorithm (QAOA) . . . . . . 5
2.3 Recursive QAOA (RQAOA) . . . . . . . . . . . . . . . . . . . . . . 7
3 Variants of RQAOA 9
3.1 RQAOA1 inspired classical algorithm (RICA) . . . . . . . . . . . . 9
3.2 Randomized RQAOA (RRQAOA) . . . . . . . . . . . . . . . . . . . 13
4 Conclusion 18
Bibliography 19

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