  A topological order is a new quantum phase that is beyond Landau's
symmetry-breaking paradigm. Its defining features include robust degenerate
ground states, long-range entanglement and anyons. It was known that $R$- and
$F$-matrices, which characterize the fusion-braiding properties of anyons, can
be used to uniquely identify topological order. In this article, we explore an
essential question: how can the $R$- and $F$-matrices be experimentally
measured? By using quantum simulations based on a toric code model with
boundaries and state-of-the-art technology, we show that the braidings, i.e.
the $R$-matrices, can be completely determined by the half braidings of
boundary excitations due to the boundary-bulk duality and the anyon
condensation. The $F$-matrices can also be measured in a scattering quantum
circuit involving the fusion of three anyons in two different orders. Thus we
provide an experimental protocol for measuring the unique identifiers of
topological order.
