发表机构
Google Quantum AI（谷歌量子人工智能）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究提出局部向量算法改进经典Max-$k$-Cut求解，并利用张量网络与qudit-玻色子等价性分析QAOA，证明在深度$p\geq9$（围长$g\geq20$）时QAOA割分数优于经典算法，展示量子优势。
AI 中文摘要
将量子优化算法的研究从二元字母表扩展到$k$元字母表已被证明为潜在的量子优势开辟了新途径。近似优化的量子优势声明要求证明，在相同假设下，高效量子算法可证明地达到比已知最佳高效经典算法所能证明的更好的性能。我们研究了在围长为$g$的$d$-正则图上的Max-$k$-Cut问题的局部经典和量子算法。我们通过基于Thompson、Parekh和Marwaha（TPM）的显式向量构造，开发了一种局部向量算法，从而推进了该问题的经典算法。对于$k\geq3$，我们的算法在已知高效经典算法中，在正则图上给出了可证明的最佳割分数保证。为了在相同的围长和正则性假设下评估QAOA的性能，我们开发了适用于一般$d$和$k\geq2$的张量网络技术。此外，利用与耦合的qudit-玻色子系统的等价性，我们计算了无限度极限下的QAOA性能。这些技术共同为大规模图上的QAOA性能提供了可证明的保证。尽管我们对经典算法进行了改进，但在深度$p\geq9$（对应围长$g\geq20$）时，QAOA在有限度和无限度情况下均实现了更好的割分数保证。因此，我们将QAOA应用于Max-$k$-Cut问题获得了明显的量子优势。
英文摘要
Broadening the study of quantum optimization algorithms from binary to $k$-element alphabets has been shown to open new avenues for potential quantum advantage. A quantum advantage claim for approximate optimization requires showing that, under the same assumptions, an efficient quantum algorithm provably achieves a better performance than can be proven for the best known efficient classical algorithms. We study local classical and quantum algorithms for Max-$k$-Cut on $d$-regular graphs of girth $g$. We advance classical algorithms for this problem by developing a local vector algorithm based on the explicit vector construction of Thompson, Parekh, and Marwaha (TPM). Our algorithm gives the best provable cut fraction guarantee among known efficient classical algorithms on regular graphs for $k\geq3$. To evaluate the performance of QAOA under identical assumptions of girth and regularity, we develop tensor network techniques for general $d$ and $k \geq 2$. In addition, using an equivalence to a coupled qudit--boson system, we compute the QAOA performance in the infinite-degree limit. Together, these techniques give provable guarantees on QAOA performance on large graphs. Despite the improvements we introduce to the classical algorithm, QAOA achieves a better cut fraction guarantee for depths $p\geq 9$, corresponding to girth $g\geq 20$, for both finite- and infinite-degree regimes. Thus, we obtain an apparent quantum advantage from applying QAOA to the Max-$k$-Cut problem.
