Quantum walk (QW), which is considered as the quantum counterpart of the classical random walk (CRW), is actually the quantum extension of CRW from the single-coin interpretation. The sequential unitary evolution engenders correlation between different steps in QW and leads to a non-binomial position distribution. In this paper, we propose an alternative quantum extension of CRW from the ensemble interpretation, named quantum random walk (QRW), where the walker has many unrelated coins, modeled as two-level systems, initially prepared in the same state. We calculate the walker’s position distribution in QRW for different initial coin states with the coin operator chosen as Hadamard matrix. In one-dimensional case, the walker’s position is the asymmetric binomial distribution. We further demonstrate that in QRW, coherence leads the walker to perform directional movement. For an initially decoherenced coin state, the walker’s position distribution is exactly the same as that of CRW. Moreover, we study QRW in 2D lattice, where the coherence plays a more diversified role in the walker’s position distribution.

中文翻译：

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相干引起的定向量子随机游动

量子步态（QW）被认为是经典随机步态（CRW）的量子对应物，实际上是单币解释中CRW的量子扩展。连续unit演化导致QW中不同步骤之间的相关性，并导致非二项式位置分布。在本文中，我们从整体解释中提出了CRW的另一种量子扩展，称为量子随机游走（QRW），其中该游走者具有许多不相关的硬币，被建模为两级系统，最初是在相同状态下制备的。我们选择硬币操作员作为Hadamard矩阵，以计算不同初始硬币状态下步行者在QRW中的位置分布。在一维情况下，步行者的位置是不对称的二项式分布。我们进一步证明，在QRW中，连贯性导致步行者进行定向运动。对于最初退相干的硬币状态，步行者的位置分布与CRW的位置分布完全相同。此外，我们在二维格子中研究QRW，其中相干性在步行者的位置分布中起着更加多样化的作用。