Laziness of Quantum Walks on Graphs

2026-08-21

AI Summary

本文研究了图上量子游走的惰性问题，通过平均混合矩阵的迹来衡量，并使用拉普拉斯量子游走工具确定了几类最惰性的连通图和树。

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.

Problem

• Quantum Walks
• Graphs
• Laziness
• Mixing Matrix
• Complete Graph

Innovation

• quantum walk
• average mixing matrix
• graph invariants
• laziness