Crossovers induced by discrete-time quantum walks
Crossovers induced by discrete-time quantum walks
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DOI：
10.26421/qic11.9-10-2
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发表时间：
2010-09
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影响因子：
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通讯作者：
Kota Chisaki;N. Konno;E. Segawa;Yutaka Shikano
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作者：
Kota Chisaki;N. Konno;E. Segawa;Yutaka Shikano
我们考虑离散时间量子行走 (DTQW) 弱收敛定理的交叉。我们证明，连续时间量子游走（CTQW）以及离散和连续时间随机游走在某些限制下可以表示为 DTQW。首先，我们概括了我们之前的研究[Phys。 Rev. A \textbf{81}, 062129 (2010)] 关于带有位置测量的 DTQW。我们表明，可以以 $p \sim 1/n^\beta$ 的概率评估每个步骤的位置测量，其中 $n$ 是最终时间，$0<\beta<1$。我们还给出了相应的连续时间情况。因此，从扩散扩散（随机游走）到弹道扩散（量子游走）的交叉可以看作是弱收敛定理的离散时间和连续时间情况下参数 $\beta$ 从 0 变为 1。其次，我们引入一类新的DTQW，其中量子币的对角线部分的绝对值与最终时间$n$的倒数的幂成正比。这称为最终时间相关 DTQW (FTD-DTQW)。 CTQW是在FTD-DTQW的极限下获得的。我们还获得了 FTD-DTQW 的弱收敛定理，该定理显示了多种扩展特性。最后，我们考虑具有周期性位置测量的 FTD-DTQW。这个弱收敛定理给出了一个相图，它映射了离散和连续时间量子和随机游走的足够长的时间行为。
We consider crossovers with respect to the weak convergence theorems from a discrete-time quantum walk (DTQW). We show that a continuous-time quantum walk (CTQW) and discrete- and continuous-time random walks can be expressed as DTQWs in some limit. At first we generalize our previous study [Phys. Rev. A \textbf{81}, 062129 (2010)] on the DTQW with position measurements. We show that the position measurements per each step with probability $p \sim 1/n^\beta$ can be evaluated, where $n$ is the final time and $0<\beta<1$. We also give a corresponding continuous-time case. As a consequence, crossovers from the diffusive spreading (random walk) to the ballistic spreading (quantum walk) can be seen as the parameter $\beta$ shifts from 0 to 1 in both discrete- and continuous-time cases of the weak convergence theorems. Secondly, we introduce a new class of the DTQW, in which the absolute value of the diagonal parts of the quantum coin is proportional to a power of the inverse of the final time $n$. This is called a final-time-dependent DTQW (FTD-DTQW). The CTQW is obtained in a limit of the FTD-DTQW. We also obtain the weak convergence theorem for the FTD-DTQW which shows a variety of spreading properties. Finally, we consider the FTD-DTQW with periodic position measurements. This weak convergence theorem gives a phase diagram which maps sufficiently long-time behaviors of the discrete- and continuous-time quantum and random walks.