Quantum Ultra-Walks: Walks on a Line with Hierarchical Spatial Heterogeneity

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 2^{rm nd}-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 identical 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.