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## "Mapping between Morita-equivalent string-net states with a constant depth quantum circuit", Laurens Lootens et al., 2022

*Reviewed December 29, 2023*

*Citation:* Lootens, Laurens, et al. "Mapping between Morita-equivalent string-net states with a constant depth quantum circuit." Physical Review B 105.8 (2022): 085130.

*Web:* https://arxiv.org/abs/2112.12757

*Tags:* Error-correcting-codes, Kiteav-quantum-double

This paper solves an interesting open problem, which generalizes several other constructions in the literature.
Namely, it shows that if D1 and D2 are unitary fusion categories such that Z(D1)\cong Z(D2), then the Levin-Wen
model corresponding to D1 is in the same topological phase as the Levin-Wen model corresponding to D2. This
condition on unitary fusion categories is known as Morita equivalence.

This is a special case of the more general conjecture that TQFTs should be in bijection with
equivalence classes of gapped Hamiltonians which realize a given TQFT at low energies. The Levin-Wen model
associated to D1 and D2 are microscopically very different - they have entirely different stabilizers -
but they are described by the same MTC and so should be equivalent.
The paper constructs nice canonical local finite depth unitary circuit that goes between the two models.

The groundwork for this paper is laid out in

> Kitaev, Alexei, and Liang Kong. "Models for gapped boundaries and domain walls." Communications in Mathematical Physics 313.2 (2012): 351-373.

where it is shown that there exist an invertible domain wall between the two
Levin-Wen phases such that anyons can move through freely without condensing.
The key insight of this paper is that this implies a microscopic equivalence by local finite depth unitaries.