离散量子游走宏微观混沌动力学特征及其在伪随机序列生成中的应用

Classical chaos has been extensively used to construct pseudo-random sequences. Unfortunately, there are few fruits on how to characterize chaos at the quantum level and how to use these chaotic characteristics for the construction of pseudo-random sequences. In the previous studies, we found there exist chaos-like characteristics in a quantum computation model, i.e., discrete quantum walks but have not made an in-depth analysis of its chaotic dynamics mechanism. This project will focus on the research on the chaotic dynamic characteristics of discrete quantum walks from the macroscopic and microscopic aspects and further use these characteristics for the construction of pseudo-random sequences. By combining the methods of theoretical analysis and numerical simulations, we will make an in-depth analysis of the entanglement dynamics of discrete quantum walks on various graphs under various conditions including single walker and multiple walkers. Then we will analyze the quantum-classical correspondence between the entanglement entropy of the reduced density matrix of coins and the indexes including the bifurcation and the Lyapunov exponent. Based on the results above, we will study the method for constructing pseudo-random sequences, analyze the properties of the constructed pseudo-random sequences in terms of multiple randomness tests and various measures such as information theory, scale index analysis, recurrence plots, etc. and try to give a clue to provable security. The merits of both high nonlinearity and parallel computing make discrete quantum walks be superior to classical chaotic maps in constructing pseudo-random sequences. The expected results can provide a new method for constructing pseudo-random sequences, provide the theoretical basis for the use of discrete quantum walks in other applications and promote the convergence between information security and quantum computation.

经典混沌理论已广泛应用于构造伪随机序列，但在量子级别如何描述混沌及利用这种混沌特征构造伪随机序列的成果相对较少。我们前期研究发现一种量子计算模型——离散量子游走具有类似混沌的特性，但对其混沌动力学特性及在构造伪随机序列中的应用还未深入研究。本课题拟从宏、微观两个层面研究其混沌动力学特性，通过理论分析与数值仿真相结合，深入研究各种图上单游走者、多游走者情形下离散量子游走的纠缠动力学特性，分析硬币约化密度矩阵的纠缠熵和分岔、Lyapunov指数之间的量子-经典对应关系，基于此混沌特性构造伪随机序列，利用多种随机性检测工具及信息论、尺度指数分析、递归图等方法，对伪随机序列的性质进行分析，并给出可证明安全性思路。离散量子游走的高度非线性和并行计算的特性使其比经典混沌理论构造伪随机序列更有优势。预期研究成果可为构造伪随机序列提供新思路，为应用于其它领域提供理论基础，也促进了信息安全和量子计算的融合。

1、项目背景.经典混沌理论已广泛应用于构造伪随机序列，但在量子级别如何描述混沌及利用这种混沌特征构造伪随机序列的成果相对较少。我们前期研究发现一种量子计算模型——离散量子游走具有类似混沌的特性，但对其混沌动力学及在构造伪随机序列中的应用还未深入研究。.2、主要研究内容.本课题从宏、微观两个层面研究其混沌动力学特性，采用理论分析与数值仿真相结合的方法，深入研究各种图上单游走者、多游走者情形下离散量子游走系统的纠缠动力学特性，分析硬币约化密度矩阵的纠缠熵和相应经典系统的分岔、Lyapunov指数之间的量子、经典对应关系，基于此特性构造伪随机序列，利用多种随机性检测工具及信息论等方法，对伪随机序列的性质进行分析，给出可证明安全性思路。.3、重要结果.项目组按照计划顺利完成了研究任务，主要成果包括：研究了线、环、二维空间、超立方体以及Johnson图上离散量子游走的纠缠动力学特性；构造了多种基于不同离散量子游走模型的伪随机序列；提出了多种基于不同离散量子游走模型的量子Hash函数；分别构造了离散量子游走框架下蝶形网络和倒皇冠网络上的量子2对单播通信方案；分别构造了基于N节点线上和N节点环上离散量子游走模型的量子隐形传态方案；提出了基于离散量子游走框架的有向非循环网络上量子态传输方案。.4、关键数据.该项目的预期考核指标为“预期研究成果被SCI收录论文12篇以上，且至少4篇发表在SCI二区以上期刊，申请发明专利1～2项”。最终项目组超额完成了计划的研究内容，共发表国际期刊 SCI 论文31篇，申请发明专利1项。论文包括《Information Sciences》2篇、《SCIENCE CHINA-Physics Mechanics & Astronomy》2篇、《SCIENCE CHINA Information Sciences》1篇、《Computers&Security》1篇、《Physics Letters A》1篇、《Quantum Information Processing》9篇及其它国际期刊15篇。.5、科学意义.离散量子游走所具有的高度非线性和并行计算的特性使其比经典混沌构造伪随机序列更有优势。研究成果为构造伪随机序列提供了全新的思路，为应用于其它领域提供理论基础，也促进了信息安全和量子计算的融合。