A new type of limit theorems for the one-dimensional quantum random walk
A new type of limit theorems for the one-dimensional quantum random walk
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DOI：
10.2969/jmsj/1150287309
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发表时间：
2005-10-01
影响因子：
0.7
通讯作者：
Konno, N
中科院分区：
文献类型：
--
作者：
Konno, N
本文考虑从由2 × 2酉矩阵U确定的初始量子态W出发的一维量子无规行走X-n（φ）在时刻n的情形。本文给出了X-n（φ）的特征函数的组合表达式。该表达式阐明了它对酉矩阵U的分量和初始量子比特态φ的依赖性。因此，我们提出了一种新的量子随机游动的极限定理。与de Moivre-Laplace极限定理相反，我们的对称情形意味着X-n（phi）/n弱收敛到极限Z（phi）为n -无穷大，其中Z（phi）的密度为1/pi（1 - x（2））root 1 - 2x（2），x是（-1/root 2，1/root 2）的元素。此外，我们讨论了一些已知的模拟结果的基础上，我们的极限定理。
In this paper we consider the one-dimensional quantum random walk X-n(phi) at time n starting from initial quoit state W determined by 2 x 2 unitary matrix U. We give a combinatorial expression for the characteristic function of X-n(phi). The expression clarifies the dependence of it on components of unitary matrix U and initial qubit state phi. As a consequence, we present a new type of limit theorems for the quantum random walk. In contrast with the de Moivre-Laplace limit theorem, our symmetric case implies that X-n(phi)/n converges weakly to a limit Z(phi) as n - infinity, where Z(phi) has a density 1/pi(1 - x(2))root 1 - 2x(2) for x is an element of (-1/root 2,1/root 2). Moreover we discuss some known simulation results based on our limit theorems.