{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,6]],"date-time":"2026-04-06T10:03:41Z","timestamp":1775469821315,"version":"3.50.1"},"reference-count":51,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2022,12,5]],"date-time":"2022-12-05T00:00:00Z","timestamp":1670198400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>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 \u201cdiffusion\u201d part, as it is called in Grover\u2019s 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 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.<\/jats:p>","DOI":"10.3390\/e24121778","type":"journal-article","created":{"date-parts":[[2022,12,5]],"date-time":"2022-12-05T07:07:31Z","timestamp":1670224051000},"page":"1778","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Quantum Spatial Search with Electric Potential: Long-Time Dynamics and Robustness to Noise"],"prefix":"10.3390","volume":"24","author":[{"given":"Thibault","family":"Fredon","sequence":"first","affiliation":[{"name":"Universit\u00e9 Paris-Saclay, CNRS, ENS Paris-Saclay, INRIA, Laboratoire M\u00e9thodes Formelles, 91190 Gif-sur-Yvette, France"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Julien","family":"Zylberman","sequence":"additional","affiliation":[{"name":"Sorbonne Universit\u00e9, Observatoire de Paris, Universit\u00e9 PSL, CNRS, LERMA, 75005 Paris, France"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Pablo","family":"Arnault","sequence":"additional","affiliation":[{"name":"Universit\u00e9 Paris-Saclay, CNRS, ENS Paris-Saclay, INRIA, Laboratoire M\u00e9thodes Formelles, 91190 Gif-sur-Yvette, France"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Fabrice","family":"Debbasch","sequence":"additional","affiliation":[{"name":"Sorbonne Universit\u00e9, Observatoire de Paris, Universit\u00e9 PSL, CNRS, LERMA, 75005 Paris, France"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,12,5]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"885","DOI":"10.1007\/s11047-019-09762-6","article-title":"An overview of quantum cellular automata","volume":"18","author":"Arrighi","year":"2019","journal-title":"Nat. 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