In this paper, we study discrete-time quantum walks on one-dimensionallattices. We find that the coherent dynamics depends on the initial states andcoin parameters. For infinite size of lattice, we derive an explicit expressionfor the return probability, which shows scaling behavior $P(0,t)\sim t^{-1}$and does not depends on the initial states of the walk. In the long-time limit,the probability distribution shows various patterns, depending on the initialstates, coin parameters and the lattice size. The average mixing time$M_{\epsilon}$ closes to the limiting probability in linear $N$ (size of thelattice) for large values of thresholds $\epsilon$. Finally, we introduceanother kind of quantum walk on infinite or even-numbered size of lattices, andshow that the walk is equivalent to the traditional quantum walk withsymmetrical initial state and coin parameter.
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机译：在本文中，我们研究了一维晶格上的离散时间量子行走。我们发现相干动力学取决于初始状态和硬币参数。对于无限大的晶格，我们为返回概率导出一个显式表达式，该表达式显示缩放行为$ P（0，t）\ sim t ^ {-1} $，并且不依赖于游走的初始状态。在长期限制中，概率分布显示出各种模式，具体取决于初始状态，硬币参数和晶格大小。对于较大阈值$ \ epsilon $，平均混合时间$ M _ {\ epsilon} $接近线性$ N $（晶格大小）的极限概率。最后，我们介绍了另一种在无穷或偶数大小的晶格上的量子步态，并表明该步态等效于具有对称初始状态和硬币参数的传统量子步态。
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