# 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} }

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…

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