Fredon Thibault, Zylberman Julien, Arnault Pablo, Debbasch Fabrice
Université Paris-Saclay, CNRS, ENS Paris-Saclay, INRIA, Laboratoire Méthodes Formelles, 91190 Gif-sur-Yvette, France.
Sorbonne Université, Observatoire de Paris, Université PSL, CNRS, LERMA, 75005 Paris, France.
Entropy (Basel). 2022 Dec 5;24(12):1778. doi: 10.3390/e24121778.
We present various results on the scheme introduced in a previous work, which is a quantum spatial-search algorithm on a two-dimensional (2D) square spatial grid, realized with a 2D Dirac discrete-time quantum walk (DQW) coupled to a Coulomb electric field centered on the the node to be found. In such a walk, the electric term acts as the oracle of the algorithm, and the free walk (i.e., without electric term) acts as the "diffusion" part, as it is called in Grover's algorithm. The results are the following. First, we run long time simulations of this electric Dirac DQW, and observe that there is a second localization peak around the node marked by the oracle, reached in a time O(N), where is the number of nodes of the 2D grid, with a localization probability scaling as O(1/lnN). This matches the state-of-the-art 2D-DQW search algorithms before amplitude amplification We then study the effect of adding noise on the Coulomb potential, and observe that the walk, especially the second localization peak, is highly robust to spatial noise, more modestly robust to spatiotemporal noise, and that the first localization peak is even highly robust to spatiotemporal noise.
我们展示了关于先前工作中引入的方案的各种结果，该方案是二维（2D）正方形空间网格上的量子空间搜索算法，通过耦合到以要寻找的节点为中心的库仑电场的二维狄拉克离散时间量子行走（DQW）实现。在这样的行走中，电项充当算法的预言机，自由行走（即没有电项）充当格罗弗算法中所谓的“扩散”部分。结果如下。首先，我们对这种电狄拉克DQW进行长时间模拟，观察到在预言机标记的节点周围存在第二个局域化峰值，在时间O(N)内达到，其中N是二维网格的节点数，局域化概率按O(1/lnN)缩放。这与幅度放大之前的最先进二维DQW搜索算法相匹配。然后我们研究了在库仑势上添加噪声的影响，观察到行走，特别是第二个局域化峰值，对空间噪声具有高度鲁棒性，对时空噪声的鲁棒性适中，并且第一个局域化峰值对时空噪声甚至具有高度鲁棒性。
