Quantum Spatial Search with Electric Potential: Long-Time Dynamics and Robustness to Noise

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(root N), where N 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.

Automated summary

This article presents results on a quantum spatial-search algorithm, implemented on a 2D square grid using a 2D Dirac discrete-time quantum walk coupled to a Coulomb electric field. The research findings show that with the addition of the electric term, the algorithm is able to reach a second localization peak around the marked node in a time of O(root N). The study also explores the effects of noise on the Coulomb potential and finds that the walk is highly robust to spatial noise, moderately robust to spatiotemporal noise, and the first localization peak is even highly robust to spatiotemporal noise.