{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2024,3,16]],"date-time":"2024-03-16T01:36:20Z","timestamp":1710552980598},"reference-count":0,"publisher":"Rinton Press","issue":"15&16","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["QIC"],"published-print":{"date-parts":[[2020,12]]},"abstract":"<jats:p>We study the localization and  the discrete probability function of a quantum search on the one-dimensional (1D) cycle with self-loops for n vertices and  m marked vertices. First, unmarked vertices have no localization since the quantum search on unmarked vertices behaves like the 1D three-state quantum walk (3QW) and localization does not occur with nonlocal initial states on a 3QW, according to residue calculations and the Riemann-Lebesgue theorem. Second, we show that localization does occur on the marked vertices and derive an analytic expression for localization by the degenerate 1-eigenvalues contributing to marked vertices. Therefore localization can contribute to a quantum search. Furthermore, we emphasize that localization comes from the self-loops. Third, using the localization of a quantum search, the asymptotic average probability distribution (AAPD)  and the discrete probability function (DPF) of a quantum search are obtained. The DPF shows that Szegedy\u2019s quantum search on the 1D cycle with self-loops spreads ballistically.<\/jats:p>","DOI":"10.26421\/qic20.15-16-2","type":"journal-article","created":{"date-parts":[[2021,2,10]],"date-time":"2021-02-10T19:57:14Z","timestamp":1612987034000},"page":"1281-1303","source":"Crossref","is-referenced-by-count":1,"title":["Localization and discrete probability function of Szegedy\u2019s quantum search one-dimensional cycle with self-loops"],"prefix":"10.26421","volume":"20","author":[{"given":"Mengke","family":"Xu","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Zhihao","family":"Liu","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Hanwu","family":"Chen","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Sihao","family":"Zheng","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"10955","published-online":{"date-parts":[[2020,12]]},"container-title":["quantum Information and Computation"],"original-title":[],"deposited":{"date-parts":[[2021,2,10]],"date-time":"2021-02-10T19:57:16Z","timestamp":1612987036000},"score":1,"resource":{"primary":{"URL":"http:\/\/www.rintonpress.com\/journals\/doi\/QIC20.15-16-2.html"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,12]]},"references-count":0,"journal-issue":{"issue":"15&16","published-online":{"date-parts":[[2020,12]]},"published-print":{"date-parts":[[2020,12]]}},"URL":"https:\/\/doi.org\/10.26421\/qic20.15-16-2","relation":{},"ISSN":["1533-7146","1533-7146"],"issn-type":[{"value":"1533-7146","type":"print"},{"value":"1533-7146","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,12]]}}}