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## "Adiabatic paths of Hamiltonians, symmetries of topological order, and automorphism codes", David Aasen, Zhenghan Wang, Matthew Hastings, 2022

*Reviewed January 1, 2024*

*Citation:* Aasen, David, Zhenghan Wang, and Matthew B. Hastings. "Adiabatic paths of Hamiltonians, symmetries of topological order, and automorphism codes." Physical Review B 106.8 (2022): 085122.

*Web:* https://arxiv.org/abs/2203.11137

*Tags:* Error-correcting-codes, Toric-code, Defects/boundaries, Floquet-theory

This paper seeks to give a general framework in which to
understand the Hastings-Haah Floquet code. The framework is as follows.
In the Hastings-Haah code, the changing of stabilizer measurements corresponds to
a changing of Hamiltonian. In this way, the Hastings-Haah code is tracing a path through
the space of gapped Hamiltonians realizing the doubled toric code phase. The fact that this
can be used to perform non-trivial computations comes from the fact that this path is
homotopically non-trivial in this moduli space.

To make this rigorous, it is important to establish that the Hastings-Haah code is not just
a sequence of stabilizers. The stabilizers in the sequence can be continuously connected to each other,
so that the discrete sequence becomes a continuos path in the moduli space. In general,
they argue that paths in the moduli space of gapped Hamiltonians should correspond
to automorphisms of MTCs. The specific path taken by the Hastings-Haah code corresponds to
charge-flux symmetry.

In this paper, the authors also consider a code they call the "e/m automorphism code".
This code forms an error correcting code at every time step, and hence is not
only proceed by Floquet cycles (like the Hastings-Haah code). The e/m automorphism
is implemented in a series of three steps, where the Hamiltonian is switched around
and a Krammers-Wannier duality circuit is run.

This paper contains a lot of good discussion of invertible domain walls, symmetries,
and gapped Hamiltonian schemas.