Three-dimensional subsystem codes from two-dimensional topological codes*

Fault-tolerant protocols and quantum error correction (QEC) are essential to building reliable quantum computers from imperfect components that are vulnerable to errors. Optimizing the resource and time overheads needed to implement QEC is currently a pressing challenge. Certain quantum error correcting codes have the single-shot error correction property, i.e., one round of parity-check measurements suffices to perform reliable QEC even in the presence of measurement errors. Three-dimensional (3D) topological subsystem codes are one class of codes with the single-shot error correction property. Examples from this class include the gauge color code and the 3D subsystem toric code. 

We provide a new perspective on 3D topological subsystem codes in terms of 2D topological codes. Our perspective clarifies the origin of the Gauss law for gauge flux, and through this the origin of the single-shot property. Furthermore, our perspective shows how the 3D subsystem toric code can be naturally generalized to Abelian quantum double models, and suggests the possibility of further generalization. We numerically investigate the perfomance of the generalized 3D subystem toric codes, finding evidence that the single-shot property is present in the entire family. We discuss Hamiltonian models for the 3D subsystem toric code, and argue that the most natural Hamiltonians are gapless.