arXiv quant-ph/06062412006-06-29quant-ph

Krylov子空间-Lanczos算法研究连续时间量子行走

Investigation of continuous-time quantum walk by using Krylov subspace-Lanczos algorithm

M. A. Jafarizadeh, S. Salimi, R. Sufiani

备注：: 29 pages, 4 figures
DOI：: 10.1140/epjb/e2007-00281-5

摘要：

在文献\cite{js,jsa}中，我们利用一种新的方法计算了具有量子分解的图（QD图）上连续时间量子行走的振幅。本文证明了任意图上的连续时间量子行走都可以用谱分布方法来研究，只需利用Krylov子空间-Lanczos算法生成量子行走的Hilbert空间的正交基同构于正交多项式.本文还引入了一类具有广义量子分解的新的图，这是通过放松对QD图的某些约束而实现的，并证明了在QD图和GQD图中，层的单位向量与由Lanczos算法产生的标准正交基是一致的。此外，还证明了在给定顶点处观察行走的概率幅与其在相应的层单位向量中的系数成正比，并可用层的幅表示.最后讨论了基于Lanczos的算法在任意图上的行走计算能力（GQD或非QD类型），已经通过计算量子行走的概率幅进行了测试，在一些有趣的有限GQD型有限（无限）图非GQD型的（无限）路图，其中概率幅在无穷多个顶点处的渐近行为，与文献\cite{nko}的中心极限定理的结果一致。

英文摘要：

In papers\cite{js,jsa}, the amplitudes of continuous-time quantum walk on graphs possessing quantum decomposition (QD graphs) have been calculated by a new method based on spectral distribution associated to their adjacency matrix. Here in this paper, it is shown that the continuous-time quantum walk on any arbitrary graph can be investigated by spectral distribution method, simply by using Krylov subspace-Lanczos algorithm to generate orthonormal bases of Hilbert space of quantum walk isomorphic to orthogonal polynomials. Also new type of graphs possessing generalized quantum decomposition have been introduced, where this is achieved simply by relaxing some of the constrains imposed on QD graphs and it is shown that both in QD and GQD graphs, the unit vectors of strata are identical with the orthonormal basis produced by Lanczos algorithm. Moreover, it is shown that probability amplitude of observing walk at a given vertex is proportional to its coefficient in the corresponding unit vector of its stratum, and it can be written in terms of the amplitude of its stratum. Finally the capability of Lanczos-based algorithm for evaluation of walk on arbitrary graphs (GQD or non-QD types), has been tested by calculating the probability amplitudes of quantum walk on some interesting finite (infinite) graph of GQD type and finite (infinite) path graph of non-GQD type, where the asymptotic behavior of the probability amplitudes at infinite limit of number of vertices, are in agreement with those of central limit theorem of Ref.\cite{nko}.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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