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

TL;DR

This paper constructs a constant depth quantum circuit that maps between Morita equivalent string-net models, demonstrating they are in the same topological phase by acting unitarily on the entire Hilbert space.

Contribution

It introduces a novel constant depth quantum circuit based on invertible bimodule categories that unitarily maps between Morita equivalent string-net models.

Findings

The circuit preserves topological order due to its constant depth and unitarity.

It acts as a generalized Fourier transform for fusion categories.

The circuit operates on the full Hilbert space with ancillas.

Abstract

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.