Connectivity is a fundamental structural property of matroids, and has been studied algorithmically over 50 years. In 1974, Cunningham proposed a deterministic algorithm consuming $O(n^{2})$ queries to the independence oracle to determine whether a matroid is connected. Since then, no algorithm, not even a random one, has worked better. To the best of our knowledge, the classical query complexity lower bound and the quantum complexity for this problem have not been considered. Thus, in this paper we are devoted to addressing these issues, and our contributions are threefold as follows: (i) First, we prove that the randomized query complexity of determining whether a matroid is connected is $\Omega(n^2)$ and thus the algorithm proposed by Cunningham is optimal in classical computing. (ii) Second, we present a quantum algorithm with $O(n^{3/2})$ queries, which exhibits provable quantum speedups over classical ones. (iii) Third, we prove that any quantum algorithm requires $\Omega(n)$ queries, which indicates that quantum algorithms can achieve at most a quadratic speedup over classical ones. Therefore, we have a relatively comprehensive understanding of the potential of quantum computing in determining the connectedness of matroids.\

翻译：连通性是拟阵的基本结构性质，其算法研究已有逾50年历史。1974年，Cunningham提出一种确定性算法，通过向独立预言机进行$O(n^{2})$次查询即可判定拟阵是否连通。此后至今，即便随机算法也未能取得更优表现。据我们所知，该问题的经典查询复杂度下界与量子复杂度此前尚未被研究。为此，本文致力于解决这些问题，主要贡献如下：(i) 首先，我们证明判定拟阵连通性的随机查询复杂度为$\Omega(n^2)$，从而Cunningham算法在经典计算中已达到最优；(ii) 其次，我们提出一种仅需$O(n^{3/2})$次查询的量子算法，证明了相对于经典算法的量子加速；(iii) 最后，我们证明任何量子算法至少需要$\Omega(n)$次查询，这表明量子算法最多能实现相对于经典算法的二次加速。由此，我们得以较全面地理解量子计算在判定拟阵连通性方面的潜力。