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

• URL: http://arxiv.org/abs/2411.16676v2
• Date: Wed, 18 Dec 2024 18:54:59 GMT
• Title: Discrete Quantum Walks with Marked Vertices and Their Average Vertex Mixing Matrices
• Authors: Amulya Mohan, Hanmeng Zhan,
• Abstract summary: We find bounds for entries in $AMM$, and study when these bounds are tight. For quantum walks attaining this bound, we determine when $AMMS,S]$ is symmetric, positive semidefinite or uniform.
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• 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.

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