Quantum Ultra-Walks: Walks on a Line with Spatial Disorder

We discuss the model of a heterogeneous discrete-time walk on a line with spatial disorder in the form of a set of ultrametric barriers. Simulations show that such an quantum ultra-walk spreads with a walk exponent $d_w$ that ranges from ballistic ($d_w=1$) to complete confinement ($d_w=\infty$) for increasing separation $1\leq1/\epsilon<\infty$ in barrier heights. We develop a formalism by which the classical random walk as well as the quantum walk can be treated in parallel 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 know as an (anti-)persistent walk. The exact analysis, based on the real-space renormalization group (RG), reproduces the results of the well-known model\footnote{J.~Phys.~A {\bf 19}(1986)L269} of ``ultradiffusion,'' $d_w=1-\log_2\epsilon$ for $0<\epsilon\leq1/2$ . However, while the evaluation of the RG fixed-points proceeds virtually identical, for the corresponding quantum walk with a unitary coin\footnote{\urllink{Phys.~Rev.~A {\bf 90}(2014)032324, \hfill http://arxiv.org/abs/1311.3369}{http://arxiv.org/abs/1311.3369}} it fails to reproduce the numerical results. A new way to analyze the RG is indicated.