This study examines clusterability testing for a signed graph in the bounded-degree model. Our contributions are two-fold. First, we provide a quantum algorithm with query complexity $\tilde{O}(N^{1/3})$ for testing clusterability, which yields a polynomial speedup over the best classical clusterability tester known [arXiv:2102.07587]. Second, we prove an $\tilde{\Omega}(\sqrt{N})$ classical query lower bound for testing clusterability, which nearly matches the upper bound from [arXiv:2102.07587]. This settles the classical query complexity of clusterability testing, and it shows that our quantum algorithm has an advantage over any classical algorithm.

翻译：本研究探讨了在有界度模型中符号图聚类性的测试问题。我们的贡献有两方面：首先，我们提出了一种查询复杂度为$\tilde{O}(N^{1/3})$的量子算法用于测试聚类性，相较于已知最优经典聚类性测试器[arXiv:2102.07587]实现了多项式加速；其次，我们证明了测试聚类性的经典查询下界为$\tilde{\Omega}(\sqrt{N})$，与[arXiv:2102.07587]的上界几乎匹配。这一结果确立了聚类性测试的经典查询复杂度，并表明我们的量子算法相较于任何经典算法具有优势。