Discrete Quantum Walks with Marked Vertices and Their Average Vertex Mixing Matrices
Amulya Mohan, Hanmeng Zhan
TL;DR
This work ties the long-time behavior of a discrete quantum walk with marked vertices to the spectral structure of the underlying graph and its subgraphs. By decomposing the walk operator $U$ into reflections and leveraging the spectrum of $X\setminus S$, the authors derive an explicit, block-wise expression for the average vertex mixing matrix $\widehat{M}$ and connect its bounds to walk-equitable neighborhoods via Schur-Complement techniques. They introduce the notion of walk-equitable collections, prove several tight bounds for $\widehat{M}[S,S]$ and $\widehat{M}[\overline{S},\overline{S}]$, and characterize when the marked-blocks are symmetric, positive semidefinite, or uniform, revealing deep links between graph structure and quantum walk mixing. The results provide a framework for analyzing return probabilities, state transfer potential, and the effect of marking on spreading behavior in regular graphs, with potential implications for quantum search and communication protocols.
Abstract
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.