  Non-Abelian topological orders offer an intriguing path towards
fault-tolerant quantum computation, where information can be encoded and
manipulated in a topologically protected manner immune to arbitrary local
noises and perturbations. However, realizing non-Abelian topologically ordered
states is notoriously challenging in both condensed matter and programmable
quantum systems, and it was not until recently that signatures of non-Abelian
statistics were observed through digital quantum simulation approaches. Despite
these exciting progresses, none of them has demonstrated the appropriate type
of topological orders and associated non-Abelian anyons whose braidings alone
support universal quantum computation. Here, we report the realization of
non-Abelian topologically ordered states of the Fibonacci string-net model and
demonstrate braidings of Fibonacci anyons featuring universal computational
power, with a superconducting quantum processor. We exploit efficient quantum
circuits to prepare the desired states and verify their nontrivial topological
nature by measuring the topological entanglement entropy. In addition, we
create two pairs of Fibonacci anyons and demonstrate their fusion rule and
non-Abelian braiding statistics by applying unitary gates on the underlying
physical qubits. Our results establish a versatile digital approach to
exploring exotic non-Abelian topological states and their associated braiding
statistics with current noisy intermediate-scale quantum processors.
