We show that, for every $k\geq 2$, $C_{2k}$-freeness can be decided in $O(n^{1-1/k})$ rounds in the \CONGEST{} model by a randomized Monte-Carlo distributed algorithm with one-sided error probability $1/3$. This matches the best round-complexities of previously known algorithms for $k\in\{2,3,4,5\}$ by Drucker et al. [PODC'14] and Censor-Hillel et al. [DISC'20], but improves the complexities of the known algorithms for $k>5$ by Eden et al. [DISC'19], which were essentially of the form $\tilde O(n^{1-2/k^2})$. Our algorithm uses colored BFS-explorations with threshold, but with an original \emph{global} approach that enables to overcome a recent impossibility result by Fraigniaud et al. [SIROCCO'23] about using colored BFS-exploration with \emph{local} threshold for detecting cycles. We also show how to quantize our algorithm for achieving a round-complexity $\tilde O(n^{\frac{1}{2}-\frac{1}{2k}})$ in the quantum setting for deciding $C_{2k}$ freeness. Furthermore, this allows us to improve the known quantum complexities of the simpler problem of detecting cycles of length \emph{at most}~$2k$ by van Apeldoorn and de Vos [PODC'22]. Our quantization is in two steps. First, the congestion of our randomized algorithm is reduced, to the cost of reducing its success probability too. Second, the success probability is boosted using a new quantum framework derived from sequential algorithms, namely Monte-Carlo quantum amplification.

翻译：我们证明，对于任意$k\geq 2$，在\CONGEST{}模型中，可通过单侧错误概率为$1/3$的随机化蒙特卡洛分布式算法，在$O(n^{1-1/k})$轮内判定$C_{2k}$-无环性。该结果与Drucker等人[PODC'14]及Censor-Hillel等人[DISC'20]针对$k\in\{2,3,4,5\}$已知算法的最优轮复杂度相匹配，但改进了Eden等人[DISC'19]针对$k>5$已知算法的复杂度（其本质上形如$\tilde O(n^{1-2/k^2})$）。我们的算法采用带阈值的彩色BFS探索，但原创性地引入\emph{全局}方法，从而克服了Fraigniaud等人[SIROCCO'23]近期关于使用\emph{局部}阈值彩色BFS探索检测环路的不可行性结果。我们还展示了如何对算法进行量子化，在量子环境下以$\tilde O(n^{\frac{1}{2}-\frac{1}{2k}})$的轮复杂度判定$C_{2k}$无环性。此外，这使我们得以改进van Apeldoorn与de Vos[PODC'22]关于检测长度\emph{至多}$~2k$环路的简单问题的已知量子复杂度。我们的量子化分两步实现：首先，降低随机化算法的拥塞程度，但代价是降低其成功概率；其次，利用源于序列算法的新型量子框架——即蒙特卡洛量子放大——提升成功概率。