  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.
