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

Understand

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.