Quantum ultra-walks: Walks on a line with hierarchical spatial heterogeneity

Abstract

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}=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}=∞) for diverging barriers. Yet for any d_{w}<∞ the quantum ultra-walk never appears to localize.