Squeezing anyons for braiding on small lattices

TL;DR

This paper demonstrates that by optimizing potentials, Abelian anyons in lattice fractional quantum Hall models can be squeezed to smaller areas, enabling more efficient braiding operations with near-ideal outcomes in compact systems.

Contribution

The authors introduce a method to shape and squeeze anyons in lattice models using optimization, reducing the area needed for braiding without losing topological properties.

Findings

Squeezed anyons can be exchanged in smaller areas with outcomes close to ideal.

The process requires about five times longer duration for adiabatic exchange.

The exchange outcome remains robust against small potential modifications.

Abstract

Adiabatically exchanging anyons gives rise to topologically protected operations on the quantum state of the system, but the desired result is only achieved if the anyons are well separated, which requires a sufficiently large area. Being able to reduce the area needed for the exchange, however, would have several advantages, such as enabling a larger number of operations per area and allowing anyon exchange to be studied in smaller systems that are easier to handle. Here, we use optimization techniques to squeeze the charge distribution of Abelian anyons in lattice fractional quantum Hall models, and we show that the squeezed anyons can be exchanged within a smaller area with a close to ideal outcome. We first use a toy model consisting of a modified Laughlin trial state to show that one can shape the anyons without altering the exchange statistics under certain conditions. We then squeeze and braid anyons in the Kapit-Mueller model and an interacting Hofstadter model by adding suitable potentials. We consider a fixed system size, for which the charge distributions of the normal anyons overlap, and we find that the outcome of the exchange process is closer to the ideal value for the squeezed anyons. The time needed for the exchange is also important, and for a particular example we find that the duration needed for the process to be close to the adiabatic limit is about five times longer for the squeezed anyons when the path length is the same. Finally we show that the exchange outcome is robust with respect to small modifications of the potential away from the optimized value.