  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 \emph{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.
