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We construct a new subsystem code in three dimensions that exhibitssingle-shot error correction in a user-friendly and transparent way. As thiscode is a subsystem version of coupled toric codes, we call it the intertwinedtoric code (ITC). Although previous codes share the property of single-shoterror correction, the ITC is distinguished by its physically motivated origin,geometrically straightforward logical operators and errors, and a simple phasediagram. The code arises from 3d stabilizer toric codes in a way thatemphasizes the physical origin of the single-shot property. In particular,starting with two copies of the 3d toric code, we add check operators thatprovide for the confinement of pointlike excitations without condensing theloop excitations. Geometrically, the bare and dressed logical operators in theITC derive from logical operators in the underlying toric codes, creating aclear relationship between errors and measurement outcomes. The syndromes ofthe ITC resemble the syndromes of the single-shot code by Kubica and Vasmer,allowing us to use their decoding schemes. We also extract the phase diagramcorresponding to ITC and show that it contains the phases found in theKubica-Vasmer code. Finally, we suggest various connections to Walker-Wangmodels and measurement-based quantum computation.

Single-Shot Quantum Error Correction in Intertwined Toric Codes