We provide numerical evidence that the nonlinear searching algorithm introduced by Wong and Meyer \cite{meyer2013nonlinear}, rephrased in terms of quantum walks with effective nonlinear phase, can be extended to the finite 2-dimensional grid, keeping the same computational advantage \BHg{with} respect to the classical algorithms. For this purpose, we have considered the free lattice Hamiltonian, with linear dispersion relation introduced by Childs and Ge \cite{Childs_2014}. The numerical simulations showed that the walker finds the marked vertex in $O(N^{1/4} \log^{3/4} N) $ steps, with probability $O(1/\log N)$, for an overall complexity of $O(N^{1/4}\log^{7/4}N)$. We also proved that there exists an optimal choice of the walker parameters to avoid that the time measurement precision affects the complexity searching time of the algorithm.

中文翻译：

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在二维网格上通过非线性量子游走搜索

我们提供了数值证据，证明由 Wong 和 Meyer \cite{meyer2013nonlinear} 引入的非线性搜索算法，重新表述为具有有效非线性相位的量子游走，可以扩展到有限二维网格，保持相同的计算优势 \BHg{相对于经典算法。为此，我们考虑了自由晶格哈密顿量，其具有由 Childs 和 Ge \cite{Childs_2014} 引入的线性色散关系。数值模拟表明，walker 在 $O(N^{1/4} \log^{3/4} N) $ 步中找到标记的顶点，概率为 $O(1/\log N)$，对于$O(N^{1/4}\log^{7/4}N)$ 的整体复杂度。我们还证明了步行器参数存在一个最优选择，以避免时间测量精度影响算法的复杂度搜索时间。