  We construct a constant depth quantum circuit that maps between Morita
equivalent string-net models. Due to its constant depth and unitarity, the
circuit cannot alter the topological order, which demonstrates that Morita
equivalent string-nets are in the same phase. The circuit is constructed from
an invertible bimodule category connecting the two input fusion categories of
the relevant string-net models, acting as a generalized Fourier transform for
fusion categories. The circuit does not only act on the ground state subspace,
but acts unitarily on the full Hilbert space when supplemented with ancillas.
