We address the properties of continuous-time quantum walks with Hamiltonians of the form $\mathscr{H}=L+\lambda L^2$, being $L$ the Laplacian matrix of the underlying graph and being the perturbation $\lambda L^2$ motivated by its potential use to introduce next-nearest-neighbor hopping. We consider cycle, complete, and star graphs because paradigmatic models with low/high connectivity and/or symmetry. First, we investigate the dynamics of an initially localized walker. Then, we devote attention to estimating the perturbation parameter $\lambda$ using only a snapshot of the walker dynamics. Our analysis shows that a walker on a cycle graph is spreading ballistically independently of the perturbation, whereas on complete and star graphs one observes perturbation-dependent revivals and strong localization phenomena. Concerning the estimation of the perturbation, we determine the walker preparations and the simple graphs that maximize the Quantum Fisher Information. We also assess the performance of position measurement, which turns out to be optimal, or nearly optimal, in several situations of interest. Besides fundamental interest, our study may find applications in designing enhanced algorithms on graphs.

中文翻译：

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存在二次扰动的连续时间量子行走

我们用以下形式的哈密顿量论连续时间量子行走的性质 $\mathscr{H}=\mathrm{大号}+\lambda {\mathrm{大号}}^2$， 存在 $\mathrm{大号}$ 基础图的拉普拉斯矩阵，并且是摄动 $\lambda {\mathrm{大号}}^2$其潜在用途是引入下一个最近邻居跳跃。我们考虑循环图，完整图和星形图，因为具有低/高连通性和/或对称性的范例模型。首先，我们研究最初定位的助行器的动力学。然后，我们专注于估计摄动参数$\lambda$仅使用步行者动态快照。我们的分析表明，周期图上的助行器独立于扰动而在弹道上扩展，而在完整图和星形图上，观察到的是与扰动有关的复兴和强烈的局部现象。关于扰动的估计，我们确定助步器的准备工作和使Quantum Fisher信息最大化的简单图形。我们还评估了位置测量的性能，结果证明在某些感兴趣的情况下，该性能是最佳或接近最佳的。除了基本兴趣之外，我们的研究还可能会发现在设计图形增强算法方面的应用。