发表机构
Yokohama National University（横滨国立大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究针对加权图上的离散时间量子游走，证明在弱边小权重极限下，分量间发现概率由简化图上的连续时间波动方程渐近描述，且该方程等价于经典弹簧-质量系统运动方程。
AI 中文摘要
我们研究加权图上的离散时间量子游走，其中边割集中的每条边被赋予一个小权重 $\epsilon>0$。参数 $\epsilon$ 表示通过割边的连接强度：当 $\epsilon \to 0$ 时，这些连接消失，图分解为移除割边后得到的连通分量。我们将这些加权的割边称为弱边。我们证明，对于足够小的 $\epsilon$，在时间尺度 $t=\Theta(\epsilon^{-1/2})$ 上，各连通分量之间的发现概率渐近地由简化图上的连续时间波动方程描述，其中简化图的顶点代表连通分量，边代表连接它们的弱边。该波动方程由一个对称的加权拉普拉斯算子控制，该算子由简化图和分量中包含的弧数决定。因此，主导项的发现概率与分量的详细内部结构无关。我们进一步证明，该波动方程与简化图上经典弹簧-质量系统的牛顿运动方程具有相同的形式。
英文摘要
We study discrete-time quantum walks on weighted graphs, where every edge in an edge cut set is assigned a small weight $ε>0$. The parameter $ε$ represents the strength of the connections through the cut edges: as $ε\to 0$, these connections vanish, and the graph decomposes into the connected components obtained by removing the cut edges. We refer to these weighted cut edges as {\it weak edges}. We show that, for sufficiently small $ε$ and on the time scale $t=Θ(ε^{-1/2})$, the finding probabilities between the connected components are asymptotically described by a continuous-time wave equation on a reduced graph, whose vertices represent the connected components and whose edges represent the weak edges connecting them. The wave equation is governed by a symmetric weighted Laplacian determined by the reduced graph and the number of arcs contained in the components. Consequently, finding probabilities of leading-term are independent of the detailed internal structures of the components. We further show that this wave equation has the same form as Newton's equation of motion for a classical spring-mass system on the reduced graph.
Comments24 pages, 2 figures
