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

TL;DR

This paper investigates a quantum spatial search algorithm on a 2D grid using a Dirac quantum walk coupled with a Coulomb electric field, demonstrating long-time localization behavior and robustness to noise.

Contribution

It extends previous work by analyzing long-time dynamics and noise robustness of a Coulomb electric field-based quantum search algorithm on a 2D grid.

Findings

Second localization peak occurs at O(√N) time with probability O(1/ln N)

Walk shows high robustness to spatial noise

First localization peak is highly robust to spatiotemporal noise

Abstract

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.