This article presents a novel and succinct algorithmic framework via alternating quantum walks, unifying quantum spatial search, state transfer and uniform sampling on a large class of graphs. Using the framework, we can achieve exact uniform sampling over all vertices and perfect state transfer between any two vertices, provided that eigenvalues of Laplacian matrix of the graph are all integers. Furthermore, if the graph is vertex-transitive as well, then we can achieve deterministic quantum spatial search that finds a marked vertex with certainty. In contrast, existing quantum search algorithms generally has a certain probability of failure. Even if the graph is not vertex-transitive, such as the complete bipartite graph, we can still adjust the algorithmic framework to obtain deterministic spatial search, which thus shows the flexibility of it. Besides unifying and improving plenty of previous results, our work provides new results on more graphs. The approach is easy to use since it has a succinct formalism that depends only on the depth of the Laplacian eigenvalue set of the graph, and may shed light on the solution of more problems related to graphs.

翻译：本文提出了一种新颖而简洁的基于交替量子行走的算法框架，该框架统一了在广泛图类上的量子空间搜索、态传输和均匀采样。利用该框架，只要图的拉普拉斯矩阵特征值均为整数，我们即可在所有顶点上实现精确的均匀采样，以及在任意两顶点间实现完美的态传输。此外，若图同时是顶点传递的，我们还能实现确定性的量子空间搜索，即能以必然性找到标记顶点。相比之下，现有的量子搜索算法通常存在一定的失败概率。即使对于非顶点传递的图（如完全二分图），我们仍可通过调整算法框架来获得确定性的空间搜索，这体现了该框架的灵活性。除了统一并改进了大量先前结果外，我们的工作还在更多类型的图上提供了新的结论。该方法易于使用，因其形式简洁，仅依赖于图的拉普拉斯特征值集合的深度，并可能为更多图相关问题的解决提供启示。