Superdiffusivity of quantum walks: A Feynman sum-over-paths description

TL;DR

This paper uses a Green's function approach to analyze quantum walks, revealing how interference effects lead to superdiffusivity and offering insights for designing quantum walks with specific transport properties.

Contribution

It introduces a general Feynman sum-over-paths framework to characterize quantum walk dynamics and superdiffusivity.

Findings

Interference effects are explicitly identified.

Superdiffusivity emerges from the sum-over-paths analysis.

Potential to guide the design of quantum walks with tailored transport properties.

Abstract

Quantum walks constitute important tools in different applications, especially in quantum algorithms. To a great extent their usefulness is due to unusual diffusive features, allowing much faster spreading than their classical counterparts. Such behavior, although frequently credited to intrinsic quantum interference, usually is not completely characterized. Using a recently developed Green's function approach [Phys. Rev. A {\bf 84}, 042343 (2011)], here it is described -- in a rather general way -- the problem dynamics in terms of a true sum over paths history a la Feynman. It allows one to explicit identify interference effects and also to explain the emergence of superdiffusivity. The present analysis has the potential to help in designing quantum walks with distinct transport properties.