We discuss the model of a one-dimensional, discrete-time walk on a line with spatial heterogeneity in the form of a variable set of ultrametric barriers. Inspired by the homogeneous quantum walk on a line, we develop a formalism by which the classical ultrametric random walk as well as the quantum walk can be treated in parallel by using a “coined” walk with internal degrees of freedom. For the random walk, this amounts to a second-order Markov process with a stochastic coin, better known as an (anti-)persistent walk. When this coin varies spatially in the hierarchical manner of “ultradiffusion,” it reproduces the well-known results of that model. The exact analysis employed for obtaining the walk dimension $d_w$, based on the real-space renormalization group (RG), proceeds virtually identically for the corresponding quantum walk with a unitary coin. However, while the classical walk remains robustly diffusive ($d_w=\frac{1}{2}$) for a wide range of barrier heights, unitarity provides for a quantum walk dimension $d_w$ that varies continuously, for even the smallest amount of heterogeneity, from ballistic spreading ($d_w=1$) in the homogeneous limit to confinement ($d_w=\infty$) for diverging barriers. Yet for any $d_w<\infty$ the quantum ultra-walk never appears to localize.

中文翻译：

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量子超步：在具有分层空间异质性的直线上行走

我们讨论了具有可变异形超集屏障形式的具有空间异质性的直线上的一维离散时间行走模型。受到直线上均质量子步态的启发，我们开发了一种形式主义，通过使用内部自由度的“投币式”步态，可以并行处理经典超计量随机步态和量子步态。对于随机游走，这相当于带有随机硬币的二阶马尔可夫过程，通常被称为（反）持久游走。当该硬币以“超融合”的分层方式在空间上变化时，它再现了该模型的众所周知的结果。用于获得步行尺寸的精确分析$d_w$的基础上，实空间重整化组（RG），前进几乎相同用于与对应的量子游走酉硬币。但是，尽管经典步道仍然具有强大的扩散性（$d_w=\frac{\mathrm{1个}}{2}$）对于大范围的势垒高度，统一性提供了量子行走尺寸 $d_w$ 从弹道扩散（即使是最小的异质性）连续变化（$d_w=\mathrm{1个}$）在均质限制范围内（$d_w=\infty$）的障碍。然而对于任何$d_w<\infty$ 量子超走似乎永远不会本地化。