  Non-Abelian anyons are exotic quasiparticle excitations hosted by certain
topological phases of matter. They break the fermion-boson dichotomy and obey
non-Abelian braiding statistics: their interchanges yield unitary operations,
rather than merely a phase factor, in a space spanned by topologically
degenerate wavefunctions. They are the building blocks of topological quantum
computing. However, experimental observation of non-Abelian anyons and their
characterizing braiding statistics is notoriously challenging and has remained
elusive hitherto, in spite of various theoretical proposals. Here, we report an
experimental quantum digital simulation of projective non-Abelian anyons and
their braiding statistics with up to 68 programmable superconducting qubits
arranged on a two-dimensional lattice. By implementing the ground states of the
toric-code model with twists through quantum circuits, we demonstrate that
twists exchange electric and magnetic charges and behave as a particular type
of non-Abelian anyons, i.e., the Ising anyons. In particular, we show
experimentally that these twists follow the fusion rules and non-Abelian
braiding statistics of the Ising type, and can be explored to encode
topological logical qubits. Furthermore, we demonstrate how to implement both
single- and two-qubit logic gates through applying a sequence of elementary
Pauli gates on the underlying physical qubits. Our results demonstrate a
versatile quantum digital approach for simulating non-Abelian anyons, offering
a new lens into the study of such peculiar quasiparticles.
