Anyonic Quantum Walks

TL;DR

This paper explores the quantum walks of anyonic systems, revealing distinct behaviors between Abelian and non-Abelian anyons, with implications for topological invariants and quantum diffusion.

Contribution

It introduces a model of anyonic quantum walks, analyzing both Abelian and non-Abelian cases, and connects the walk dynamics to topological invariants like Jones polynomials.

Findings

Abelian anyonic walks show quadratic quantum speedup.

Non-Abelian anyonic walks exhibit complex behaviors due to Hilbert space growth.

Position distributions relate to topological invariants such as Jones polynomials.

Abstract

The one dimensional quantum walk of anyonic systems is presented. The anyonic walker performs braiding operations with stationary anyons of the same type ordered canonically on the line of the walk. Abelian as well as non-Abelian anyons are studied and it is shown that they have very different properties. Abelian anyonic walks demonstrate the expected quadratic quantum speedup. Non-Abelian anyonic walks are much more subtle. The exponential increase of the system's Hilbert space and the particular statistical evolution of non-Abelian anyons give a variety of new behaviors. The position distribution of the walker is related to Jones polynomials, topological invariants of the links created by the anyonic world-lines during the walk. Several examples such as the SU(2) level k and the quantum double models are considered that provide insight to the rich diffusion properties of anyons.