The quantum-walk-based spatial search problem aims to find a marked vertex using a quantum walk on a graph with marked vertices. We describe a framework for determining the computational complexity of spatial search by continuous-time quantum walk on arbitrary graphs by providing a recipe for finding the optimal running time and the success probability of the algorithm. The quantum walk is driven by a Hamiltonian derived from the adjacency matrix of the graph modified by the presence of the marked vertices. The success of our framework depends on the knowledge of the eigenvalues and eigenvectors of the adjacency matrix. The spectrum of the Hamiltonian is subsequently obtained from the roots of the determinant of a real symmetric matrix $M$, the dimensions of which depend on the number of marked vertices. The eigenvectors are determined from a basis of the kernel of $M$. We show each step of the framework by solving the spatial searching problem on the Johnson graphs with a fixed diameter and with two marked vertices. Our calculations show that the optimal running time is $O(\sqrt{N})$ with an asymptotic probability of $1+o(1)$, where $N$ is the number of vertices.

翻译：量子游走空间搜索问题旨在利用带有标记顶点的图上的量子游走来寻找一个标记顶点。我们描述了一个框架，通过提供寻找算法最优运行时间和成功概率的方法，来确定任意图上连续时间量子游走空间搜索的计算复杂度。该量子游走由基于图的邻接矩阵（经标记顶点修改后）推导出的哈密顿量驱动。我们框架的成功依赖于对邻接矩阵特征值和特征向量的了解。哈密顿量的谱随后通过一个实对称矩阵 $M$ 的行列式根获得，该矩阵的维度取决于标记顶点的数量。特征向量由 $M$ 的核基确定。我们通过求解具有固定直径和两个标记顶点的约翰逊图上的空间搜索问题，展示了该框架的每一步。我们的计算表明，最优运行时间为 $O(\sqrt{N})$，渐进成功概率为 $1+o(1)$，其中 $N$ 是顶点数量。