This work examines the time complexity of quantum search algorithms on combinatorial t-designs with multiple marked elements using the continuous-time quantum walk. Through a detailed exploration of t-designs and their incidence matrices, we identify a subset of bipartite graphs that are conducive to success compared to random-walk-based search algorithms. These graphs have adjacency matrices with eigenvalues and eigenvectors that can be determined algebraically and are also suitable for analysis in the multiple-marked vertex scenario. We show that the continuous-time quantum walk on certain symmetric t-designs achieves an optimal running time of \(O\left( \sqrt{n}\right) \), where n is the number of points and blocks, even when accounting for an arbitrary number of marked elements. Upon examining two primary configurations of marked elements distributions, we observe that the success probability is consistently o(1), but it approaches 1 asymptotically in certain scenarios.

这项工作使用连续时间量子行走检查具有多个标记元素的组合t设计的量子搜索算法的时间复杂度。通过对t设计及其关联矩阵的详细探索，我们确定了与基于随机游走的搜索算法相比有利于成功的二分图子集。这些图具有带有特征值和特征向量的邻接矩阵，可以通过代数方法确定，并且也适合在多标记顶点场景中进行分析。我们证明，某些对称t设计上的连续时间量子行走实现了\(O\left( \sqrt{n}\right) \)的最佳运行时间，其中n是点和块的数量，即使当考虑任意数量的标记元素。在检查标记元素分布的两个主要配置时，我们观察到成功概率始终为o (1)，但在某些情况下它渐近地接近 1。