We study the discrete quantum walk on a regular graph $X$ that assigns negative identity coins to marked vertices $S$ and Grover coins to the unmarked ones. We find combinatorial bases for the eigenspaces of the transtion matrix, and derive a formula for the average vertex mixing matrix $\AMM$. We then find bounds for entries in $\AMM$, and study when these bounds are tight. In particular, the average probabilities between marked vertices are lower bounded by a matrix determined by the induced subgraph $X[S]$, the vertex-deleted subgraph $X\backslash S$, and the edge deleted subgraph $X-E(S)$. We show this bound is achieved if and only if the marked vertices have walk-equitable neighborhoods in the vertex-deleted subgraph. Finally, for quantum walks attaining this bound, we determine when $\AMM[S,S]$ is symmetric, positive semidefinite or uniform.

翻译：我们研究了在正则图$X$上的离散量子行走，该行走为标记顶点$S$分配负恒等硬币，为非标记顶点分配Grover硬币。我们找到了转移矩阵特征空间的组合基，并推导出了平均顶点混合矩阵$\AMM$的公式。随后，我们给出了$\AMM$中元素的上界，并研究了这些上界何时达到紧致。特别地，标记顶点之间的平均概率由一个由诱导子图$X[S]$、顶点删除子图$X\backslash S$以及边删除子图$X-E(S)$所确定的矩阵下界。我们证明，当且仅当标记顶点在顶点删除子图中具有行走等价的邻域时，此下界可达。最后，对于达到此下界的量子行走，我们确定了$\AMM[S,S]$何时是对称的、半正定的或均匀的。