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.

中文翻译：

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哈密​​顿量的绝热路径、拓扑序的对称性和自同构码

最近的“蜂窝码”是由一系列检查定义的容错量子存储器，它实现了 toric 码的非平凡自同构。我们认为理解该代码的一般框架是考虑间隙哈密顿量的连续绝热路径，并且我们在两个空间维度上对该空间的基本群以及第二和第三同伦群进行了推测描述。这种路径的单个循环可以实现该哈密顿量的拓扑顺序的某种自同构。我们为二维双拓扑顺序的任意自同构构造这样的路径。然后，在 toric 代码的情况下实现这一点，我们将这条路径转回一系列检查，构造一个与蜂窝代码密切相关的自同构代码。