Mapping between Morita-equivalent string-net states with a constant depth quantum circuit

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