Discrete Quantum Walks with Marked Vertices and Their Average Vertex Mixing Matrices

TL;DR

This paper analyzes discrete quantum walks on regular graphs with marked vertices, deriving eigenspaces, average mixing matrices, and bounds, revealing conditions for optimal mixing and symmetry properties.

Contribution

It introduces combinatorial eigenspace bases, formulas for average mixing matrices, and bounds for entries, with conditions for tightness and symmetry in quantum walks with marked vertices.

Findings

Derived eigenspaces for transition matrices.

Established bounds for average vertex mixing matrix entries.

Identified conditions for symmetry and positivity of the mixing matrix.

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.