电子学报 ›› 2021, Vol. 49 ›› Issue (7): 1323-1330.DOI: 10.12263/DZXB.20200709

• 学术论文 • 上一篇    下一篇

一维离散时间量子行走的路径分析法、概率分布与对称性

林运国1,2   

  1. 1.福建农林大学计算机与信息学院,福建 福州 350002
    2.南威软件集团博士后科研工作站,福建 泉州 362000
  • 收稿日期:2020-07-14 修回日期:2021-01-05 出版日期:2021-07-25 发布日期:2021-08-11
  • 作者简介:林运国 男,1979年出生于福建福清,博士,副教授,主要研究方向为量子计算与量子信息、模型检测.E‑mail:linyg@fafu.edu.cn
  • 基金资助:
    福建省自然科学基金(2016J01283)

One‑Dimensional Discrete Time Quantum Walk: Path Analysis Approach, Probability Distribution and Symmetry

Yun-guo LIN1,2   

  1. 1.College of Computer and Information Sciences,Fujian Agriculture and Forestry University,Fuzhou,Fujian 350002,China
    2.Post?Doctoral Research Center,Linewell Software Co. ,Ltd. ,Quanzhou,Fujian 362000,China
  • Received:2020-07-14 Revised:2021-01-05 Online:2021-07-25 Published:2021-08-11

摘要:

为了模拟长程的一维离散时间量子行走的演化过程以及降低计算复杂性,提出一种路径分析方法.首先分析系统到达某个位置的路径分块和路径数,将系统移位到某个位置演化算子进行分解,表示成二阶矩阵线性空间的一组基的线性组合;然后用超几何级数进行化简,给出概率分布的计算方法;最后分析产生对称式概率分布的充分条件,表明对称性只与量子初态有关.实验结果表明,该方法能够有效模拟系统的长时间演化过程.相关结果可以推广到更一般类型的离散时间量子行走.

关键词: 离散时间量子行走, 翻转算子, 路径分析法, 极限分布, 概率分布, 对称性

Abstract:

In order to simulate one?dimensional discrete time quantum walks after a long time of evolution and reduce the computational complexity, we propose a path analysis approach. Dividing every path of a discrete time quantum walk into a set of blocks and counting number of paths starting from an initial qubit state, we give an expression for some evolution operator in form of a linear combination of a fixed set of basis in a 2?dimensional matrix space. As a consequence, we present a calculating formula of probability distribution in terms of hypergeometric polynomial and give a sufficient condition on symmetry of probability distribution which is only determined by initial qubit states. Our experimental results reveal that this method can well simulate the evolution process in a long time scale. We also briefly extend these studies to more general discrete time quantum walks.

Key words: discrete time quantum walk, flip operator, path analysis approach, limit distribution, probability distribution, symmetry

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