We study the computational phase transition in a multi-frequency group synchronization problem, where pairwise relative measurements of group elements are observed across multiple frequency channels and corrupted by Gaussian noise. Using the framework of \emph{low-degree polynomial algorithms}, we analyze the task of estimating the structured signal in such observations. We show that, assuming the low-degree heuristic, in synchronization models over the circle group $\mathsf{SO}(2)$, a simple spectral method is computationally optimal among all polynomial-time estimators when the number of frequencies satisfies $L=n^{o(1)}$. This significantly extends prior work \cite{KBK24+}, which only applied to a fixed constant number of frequencies. Together with known upper bounds on the statistical threshold \cite{PWBM18a}, our results establish the existence of a \emph{statistical-to-computational gap} in this model when the number of frequencies is sufficiently large.

翻译：我们研究多频群同步问题中的计算相变，其中群元素的成对相对观测通过多个频率通道获取，并受到高斯噪声干扰。利用\emph{低次多项式算法}框架，我们分析了从此类观测中估计结构化信号的任务。我们证明，在圆群$\mathsf{SO}(2)$上的同步模型中，若假设低次启发式成立，当频率数满足$L=n^{o(1)}$时，简单的谱方法在所有多项式时间估计器中是计算最优的。这显著扩展了先前工作\cite{KBK24+}（仅适用于固定常数个频率的情形）。结合已知的统计阈值上界\cite{PWBM18a}，我们的结果确立了当频率数足够大时，该模型中存在\emph{统计-计算间隙}。