论文标题
Grover Walk的电路方程
Circuit equation of Grover walk
论文作者
论文摘要
我们考虑了无限图上的Grover Walk，其中内部有限的子图从外部收到流入，并以一定的频率将流出辐射到外部。为了表征该系统的固定状态，该系统由图表上的函数表示，我们引入了一种由频率扭曲的离散梯度操作员。然后，我们获得了一个电路方程，该方程表明（i）固定状态由潜在函数的扭曲梯度描述，该函数是顶点上的函数； （ii）潜在函数满足泊松方程相对于广义拉普拉斯矩阵。因此，我们表征了内部图表面及其内部能量的散射。此外，对于完整的图作为内部图，我们说明了散射与内部能量与频率和尾巴数的关系。
We consider the Grover walk on the infinite graph in which an internal finite subgraph receives the inflow from the outside with some frequency and also radiates the outflow to the outside. To characterize the stationary state of this system, which is represented by a function on the arcs of the graph, we introduce a kind of discrete gradient operator twisted by the frequency. Then we obtain a circuit equation which shows that (i) the stationary state is described by the twisted gradient of a potential function which is a function on the vertices; (ii) the potential function satisfies the Poisson equation with respect to a generalized Laplacian matrix. Consequently, we characterize the scattering on the surface of the internal graph and the energy penetrating inside it. Moreover, for the complete graph as the internal graph, we illustrate the relationship of the scattering and the internal energy to the frequency and the number of tails.
