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Fractional scaling of quantum walks on percolation lattices

University of British Columbia. Department of Physics and Astronomy; Workshop on Quantum Algorithms, Computational Models and Foundations of Quantum Mechanics; Pacific Institute for the Mathematical Sciences; Summer School on Quantum Information (10th : 2010 : Vancouver, B.C.)

Quantum walks have been used as simple models of quantum transport phenomena, applicable to systems as diverse as spin chains and bio-molecules. 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 T to the power alpha (0.5 < alpha < 1). Our (numerical) results are skewed by finite size effects: the increase in alpha from zero begins before p = pc. It then flattens toward the classical random walk spreading rate of alpha = 0.5 around p = 0.85, followed by a steep rise to the quantum value of alpha = 1 at p = 1. At this stage, we do not have enough data to predict the large T behaviour, but think the steep rise will becomes a "step" function at p = 1 as T -> infinity. The randomness in the percolation lattice would thus act as decoherence in the large T limit. However, such a limit could only be approached for quite large values of T, and from the point of view of models for disordered systems on smaller scales (tens of sites), the faster-than-classical fractional scaling is very much the dominant feature.

Presentation

eng

Vancouver : University of British Columbia Library

10.14288/1.0103159

Non UBC

Researcher

DSpace