Universal quantum computers are potentially an ideal setting for simulating emergent quantum many-body phenomena that are out of reach for classical computers. Here we discuss two applications to the study of topologically ordered systems. First, we represent the ground states of Hamiltonians using shallow quantum circuits and observe a quantum phase transition between different symmetry-protected topological phases on a quantum device. Second, we prepare the ground state of the toric code Hamiltonian using an efficient quantum circuit on a superconducting quantum processor. We measure a topological entanglement entropy near the expected value of ln2, and simulate anyon interferometry to extract the braiding statistics of the emergent excitations.