Fractional scaling of quantum walks on percolation lattices

Quantum walks have been used as simple models of quantum transport phenomena, applicable to systems as diverse as spin chains [1] and bio-molecules [2]. Here we investigate the properties of quantum walks on percolation lattices, disordered structures appropriate for modelling biological and experimentally realistic systems. Both bond (edge) and site percolation have similar definitions: with independent randomly chosen probability p the bond or site is present in the lattice. In two and higher dimensions, percolation lattices exhibit a phase transition from a set of small disconnected regions to a more highly connected structure (“one giant cluster”). On 2D Cartesian lattices, the critical probability pc = 0.5 (bond) and pc = 0.5927... (site). Below pc, the quantum walk will not be able to spread. Approaching pc from above, the spreading slows down completely, as the number of long-distance connected paths reduces to zero. For p = 1, the lattice is fully connected, and the standard quantum walk spreading applies (linear in t). In between, we find the quantum walks show fractional scaling of the spreading, i.e., proportional to tα for (0.5 < α < 1). The numerically determined scaling exponents α are shown in fig.