{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,1]],"date-time":"2025-12-01T06:35:47Z","timestamp":1764570947323},"reference-count":0,"publisher":"Rinton Press","issue":"13&14","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["QIC"],"published-print":{"date-parts":[[2019,11]]},"abstract":"<jats:p>Dukes (2014) and Konno, Shimizu, and Takei (2017) studied the periodicity for 2-state quantum walks whose coin operator is the Hadamard matrix on cycle graph C_N with N vertices. The present paper treats the periodicity for 3-state quantum walks on C_N. Our results follow from a new method based on the cyclotomic field. This method gives a necessary condition for the coin operator for quantum walks to be periodic. Moreover, we reveal the period T_N of typical two kinds of quantum walks, the Grover and Fourier walks. We prove that both walks do not have any finite period except for N=3, in which case T_3=6 (Grover), =12 (Fourier).<\/jats:p>","DOI":"10.26421\/qic19.13-14-1","type":"journal-article","created":{"date-parts":[[2021,2,15]],"date-time":"2021-02-15T19:18:39Z","timestamp":1613416719000},"page":"1081-1088","source":"Crossref","is-referenced-by-count":3,"title":["Periodicity for the 3-state quantum walk on cycles"],"prefix":"10.26421","volume":"19","author":[{"given":"Takeshi","family":"Kajiwara","sequence":"first","affiliation":[]},{"given":"Norio","family":"Konno","sequence":"additional","affiliation":[]},{"given":"Shohei","family":"Koyama","sequence":"additional","affiliation":[]},{"given":"Kei","family":"Saito","sequence":"additional","affiliation":[]}],"member":"10955","published-online":{"date-parts":[[2019,11]]},"container-title":["quantum Information and Computation"],"original-title":[],"deposited":{"date-parts":[[2021,2,15]],"date-time":"2021-02-15T19:18:41Z","timestamp":1613416721000},"score":1,"resource":{"primary":{"URL":"http:\/\/www.rintonpress.com\/journals\/doi\/QIC19.13-14-1.html"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,11]]},"references-count":0,"journal-issue":{"issue":"13&14","published-online":{"date-parts":[[2019,11]]},"published-print":{"date-parts":[[2019,11]]}},"URL":"https:\/\/doi.org\/10.26421\/qic19.13-14-1","relation":{},"ISSN":["1533-7146","1533-7146"],"issn-type":[{"value":"1533-7146","type":"print"},{"value":"1533-7146","type":"electronic"}],"subject":[],"published":{"date-parts":[[2019,11]]}}}