  Non-Abelian topological order (TO) is a coveted state of matter with
remarkable properties, including quasiparticles that can remember the sequence
in which they are exchanged. These anyonic excitations are promising building
blocks of fault-tolerant quantum computers. However, despite extensive efforts,
non-Abelian TO and its excitations have remained elusive, unlike the simpler
quasiparticles or defects in Abelian TO. In this work, we present the first
unambiguous realization of non-Abelian TO and demonstrate control of its
anyons. Using an adaptive circuit on Quantinuum's H2 trapped-ion quantum
processor, we create the ground state wavefunction of $D_4$ TO on a kagome
lattice of 27 qubits, with fidelity per site exceeding $98.4\%$. By creating
and moving anyons along Borromean rings in spacetime, anyon interferometry
detects an intrinsically non-Abelian braiding process. Furthermore, tunneling
non-Abelions around a torus creates all 22 ground states, as well as an excited
state with a single anyon -- a peculiar feature of non-Abelian TO. This work
illustrates the counterintuitive nature of non-Abelions and enables their study
in quantum devices.
