  Direct experimental detection of anyonic exchange statistics in fractional
quantum Hall systems by braiding the excitations and measuring the
wave-function phase is an enormous challenge. Here, we use a small, noisy
quantum computer to emulate direct braiding within the framework of a
simplified model applicable to a thin cylinder geometry and measure the
topological phase. Our algorithm first prepares the ground state with two
quasiholes. It then applies a unitary operation controlled by an ancilla,
corresponding to a sequence of adiabatic evolutions that takes one quasihole
around the other. We finally extract the phase of the wave function from
measuring the ancilla with a compound error mitigation strategy. Our results
open a new avenue for studying braiding statistics in fractional Hall states.
