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The recent "honeycomb code" is a fault-tolerant quantum memory defined by asequence of checks which implements a nontrivial automorphism of the toriccode. We argue that a general framework to understand this code is to considercontinuous adiabatic paths of gapped Hamiltonians and we give a conjectureddescription of the fundamental group and second and third homotopy groups ofthis space in two spatial dimensions. A single cycle of such a path canimplement some automorphism of the topological order of that Hamiltonian. Weconstruct such paths for arbitrary automorphisms of two-dimensional doubledtopological order. Then, realizing this in the case of the toric code, we turnthis path back into a sequence of checks, constructing an automorphism codeclosely related to the honeycomb code.

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