We introduce a fidelity-based measure, ${\text{D}}_{\mathrm{QC}}\left(t\right)$, to quantify the differences in the dynamics of classical versus quantum walks over a graph. We provide universal, graph-independent, analytic expressions of this quantum-classical dynamical distance, showing that at short times ${\text{D}}_{\mathrm{QC}}\left(t\right)$ is proportional to the coherence of the walker, i.e., a genuine quantum feature, whereas at long times it depends only on the size of the graph. At intermediate times, ${\text{D}}_{\mathrm{QC}}\left(t\right)$ does depend on the graph topology through its algebraic connectivity. Our results show that the difference in the dynamical behavior of classical and quantum walks is entirely due to the emergence of quantum features at short times. In the long-time limit, quantumness and the different nature of the generators of the dynamics, e.g., the open-system nature of classical walks and the unitary nature of quantum walks, are instead contributing equally.

中文翻译：

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量子游动的量子经典动力学距离和量子性

我们引入了基于保真度的度量， ${\text{d}}_{\mathrm{质量控制}}（Ť）$，以量化图上经典游动与量子游动的动力学差异。我们提供了该量子经典动力学距离的通用，与图无关的解析表达式，表明在短时间内${\text{d}}_{\mathrm{质量控制}}（Ť）$它与助步器的相干性成正比，即一个真正的量子特征，而长期来看，它仅取决于图的大小。在中间时间${\text{d}}_{\mathrm{质量控制}}（Ť）$确实通过其代数连接性取决于图拓扑。我们的研究结果表明，经典与量子行走动力学行为的差异完全是由于短时间内出现了量子特征所致。在长期的限制中，量子性和动力学生成器的不同性质，例如经典步行的开放系统性质和量子步行的单一性质，反而同样发挥了作用。