Quantum spatial search with electric potential : long-time dynamics and robustness to noise

We present various results on the scheme introduced , 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 marked node. 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 simulations of this electric Dirac DQW during longer times than explored in Ref.\ \cite{ZD21}, and observe that there is a second localization peak around the node marked by the oracle, reached in a time $O(\sqrt{N})$, where $N$ is the number of nodes of the 2D grid, with a localization probability scaling as $O(1/\ln N)$. 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.