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arXiv · 2608.20739

Laziness of Quantum Walks on Graphs

Amulya Mohan·Christino Tamon·Yichi Xu·Hanmeng Zhan
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Abstract

The trace of the average mixing matrix of a quantum walk measures the "laziness" of the walk: the higher the trace, the more likely that the walker returns home in the long run. In this paper, we develop tools to study this graph invariant arising from Laplacian quantum walks. It is known that the complete graph $K_n$ is the laziest connected graph on $n$ vertices. Using our machinery, we show that the star $S_n$ is the second laziest connected graph on $n$ vertices (and hence the laziest tree on $n$ vertices), the complete multipartite graph $K_{n-2,1,1}$ is the third laziest connected graph on $n$ vertices, and the double star $DS(n-3,1)$ is the second laziest tree on $n$ vertices. We also show that on the same number of vertices, more unbalanced double stars are lazier.

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BibTeXRIS

Amulya Mohan, Christino Tamon, Yichi Xu, Hanmeng Zhan. 2026-08-21. Laziness of Quantum Walks on Graphs. https://arxiv.org/abs/2608.20739

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