• Corpus ID: 247594050

Adiabatic paths of Hamiltonians, symmetries of topological order, and automorphism codes

@inproceedings{Aasen2022AdiabaticPO,
  title={Adiabatic paths of Hamiltonians, symmetries of topological order, and automorphism codes},
  author={David Aasen and Zhenghan Wang and Matthew B. Hastings},
  year={2022}
}

• D. Aasen, Zhenghan Wang, M. Hastings
• Published 21 March 2022
• Mathematics

The recent “honeycomb code” is a fault-tolerant quantum memory defined by a sequence of checks which implements a nontrivial automorphism of the toric code. We argue that a general framework to understand this code is to consider continuous adiabatic paths of gapped Hamiltonians and we give a conjectured description of the fundamental group and second and third homotopy groups of this space in two spatial dimensions. A single cycle of such a path can implement some automorphism of the topological order of that Hamiltonian. We construct such paths for arbitrary automorphisms of two-dimensional doubled topological order. Then, realizing this in the case of the toric code, we turn this path back into a sequence of checks, constructing an automorphism code closely related to the honeycomb code. on all type r mod 3 plaquettes starting with r = 0. We begin by running the Krammers-Wannier circuit on all type 0 plaquettes. The type 0 plaquettes have type 1 and 2 edges at there boundaries. After running the Kramers-Wannier circuit on the type 0 plaquettes, the type