A self-correcting quantum memory can store and protect quantum information for a time that increases without bound with the system size and without the need for active error correction. We demonstrate that symmetry can lead to self-correction in 3D spin-lattice models. In particular, we investigate codes given by 2D symmetry-enriched topological (SET) phases that appear naturally on the boundary of 3D symmetry-protected topological (SPT) phases. We find that while conventional on-site symmetries are not sufficient to allow for self-correction in commuting Hamiltonian models of this form, a generalized type of symmetry known as a 1-form symmetry is enough to guarantee self-correction. We illustrate this fact with the 3D “cluster-state” model from the theory of quantum computing. This model is a self-correcting memory, where information is encoded in a 2D SET-ordered phase on the boundary that is protected by the thermally stable SPT ordering of the bulk. We also investigate the gauge color code in this context. Finally, noting that a 1-form symmetry is a very strong constraint, we argue that topologically ordered systems can possess emergent 1-form symmetries, i.e., models where the symmetry appears naturally, without needing to be enforced externally.

中文翻译：

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受对称保护的自校正量子内存

自校正量子存储器可以存储和保护量子信息一段延长的时间，而不受系统大小的限制，也不需要主动错误校正。我们证明了对称性可以导致3D自旋晶格模型中的自我校正。特别是，我们研究了2D对称性丰富的拓扑（SET）阶段给出的代码，这些代码自然出现在3D对称性保护的拓扑（SPT）阶段的边界上。我们发现，尽管常规的现场对称性不足以允许在这种形式的哈密顿通勤模型中进行自我校正，但被称为1形对称性的广义对称性类型足以保证自我校正。我们用来自量子计算理论的3D“集群状态”模型来说明这一事实。此模型是一种自我校正的记忆，其中信息以2D SET排序的相位编码在边界上，该边界受整体的热稳定SPT排序保护。我们还将在这种情况下调查量规颜色代码。最后，注意到1形式的对称性是一个非常强的约束，我们认为拓扑排序的系统可以拥有出现的1形式对称，即自然出现对称的模型，无需从外部强制执行。