Limit distribution of a time-dependent quantum walk on the half line
- PDF / 1,707,248 Bytes
- 18 Pages / 439.37 x 666.142 pts Page_size
- 50 Downloads / 165 Views
Limit distribution of a time-dependent quantum walk on the half line Takuya Machida1 Received: 4 March 2020 / Accepted: 5 August 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract We focus on a two-period time-dependent quantum walk on the half line in this paper. The quantum walker launches at the edge of the half line in a localized superposition state, and its time evolution is carried out with two unitary operations which are alternately casted to the quantum walk. As a result, long-time limit finding probabilities of the quantum walk turn to be determined by either one of the two operations, but not both. More interestingly, the limit finding probabilities are independent from the localized initial state. We will approach the appreciated features via a quantum walk on the line which is able to reproduce the time-dependent walk on the half line. Keywords Time-dependent quantum walk · Half line · Limit distribution
1 Introduction As one of the quantum counterparts of random walks, quantum walks have been investigated since around 2000 and many features of them have been discovered. The most appreciated feature is that probability distributions of the quantum walks show ballistic spread as their time evolutions are promoting. Also the probability distributions are not similar to the Gauss distributions which are known for the approximate distributions of random walks. The intriguing features have been applied for quantum algorithms and turned out to prove that the algorithms can perform quadratic speed-up [1]. In this paper, we focus on a quantum walk on the half line and attempt to get its long-time limit distributions. This study is motivated on long-time limit distributions because they describe how the quantum walkers behave after long time, and the limit distributions for time-dependent quantum walks have not been derived. The walker moves around the locations represented by the set of nonnegative integers {0, 1, 2, . . .}. Limit distributions were analyzed for several quantum walks on the half line in the past
B 1
Takuya Machida [email protected] College of Industrial Technology, Nihon University, Narashino, Chiba 275-8576, Japan 0123456789().: V,-vol
123
296
Page 2 of 18
T. Machida
studies [2–4]. While we take care of a time-dependent quantum walk in this study, the past researches on quantum walks on the half line were all for time-independent walks. Konno and Segawa [2] found long-time limit measures of two types of quantum walk on the half line. Each type had a large mass in distribution, and the mass was described as localization. The presence of localization allowed them to derive limit measures with which where the quantum walker was observed after its unitary evolution ran infinite times. On the other hand, a limit theorem on a rescaled space by time was proved by Liu and Petulante [3]. The limit theorem depicted the approximate and global shape of the probability distribution of a quantum walk on the half line. Machida [4] discovered a relation b
Data Loading...
