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.

• Received 7 April 2022
• Revised 5 July 2022
• Accepted 11 July 2022

DOI:https://doi.org/10.1103/PhysRevB.106.085122