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 $O(\sqrt{n})$, 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.

翻译：本文研究了在组合t-设计上利用连续时间量子游走进行包含多个标记元素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深入探究，我们识别出一类相较于基于随机游走的搜索算法更易取得成功的二部图子集。这些图的邻接矩阵具有可代数求解的特征值与特征向量，且适用于多标记顶点场景的分析。研究表明，在特定对称t-设计上，即使考虑任意数量的标记元素，连续时间量子游走仍能达到最优运行时间$O(\sqrt{n})$（其中$n$为点与块的数量）。通过考察两种主要的标记元素分布配置，我们观察到成功概率始终为$o(1)$，但在某些场景下会渐近趋近于1。