  Topological orders can be used as media for topological quantum computing ---
a promising quantum computation model due to its invulnerability against local
errors. Conversely, a quantum simulator, often regarded as a quantum computing
device for special purposes, also offers a way of characterizing topological
orders. Here, we show how to identify distinct topological orders via measuring
their modular $S$ and $T$ matrices. In particular, we employ a nuclear magnetic
resonance quantum simulator to study the properties of three topologically
ordered matter phases described by the string-net model with two string types,
including the $\Z_2$ toric code, doubled semion, and doubled Fibonacci. The
third one, non-Abelian Fibonacci order is notably expected to be the simplest
candidate for universal topological quantum computing. Our experiment serves as
the basic module, built on which one can simulate braiding of non-Abelian
anyons and ultimately topological quantum computation via the braiding, and
thus provides a new approach of investigating topological orders using quantum
computers.
