Braiding Interactions in Anyonic Quantum Walks

TL;DR

This paper reviews how braiding statistics influence anyonic quantum walks, showing that non-Abelian anyons induce entanglement and decoherence, with potential experimental realizations in fractional quantum Hall systems.

Contribution

It provides a comprehensive review of recent results on anyonic quantum walks, highlighting the effects of braiding and entanglement, and presents numerical analysis of mixing times.

Findings

Non-Abelian anyons cause entanglement with topological degrees of freedom.

Decoherence leads to classical-like behavior in infinite ladder walks.

Numerical results on mixing times for $SU(2)_2$ anyon walks.

Abstract

The anyonic quantum walk is a dynamical model describing a single anyon propagating along a chain of stationary anyons and interacting via mutual braiding statistics. We review the recent results on the effects of braiding statistics in anyonic quantum walks in quasi-one dimensional ladder geometries. For anyons which correspond to spin-1/2 irreps of the quantum groups $SU(2)_k$, the non-Abelian species $(1<k<\infty)$ gives rise to entanglement between the walker and topological degrees of freedom which is quantified by quantum link invariants over the trajectories of the walk. The decoherence is strong enough to reduce the walk on the infinite ladder to classical like behaviour. We also present numerical results on mixing times of $SU(2)_2$ or Ising model anyon walks on cyclic graphs. Finally, the possible experimental simulation of the anyonic quantum walk in Fractional Quantum Hall systems is discussed.