Linking quantum error correction and gauge theories with quantum reference frames

Redundancy is the hallmark of both quantum error correction and gauge theories. In this talk, I will show that this analogy is not merely a coincidence but that there is a deeper underlying structural relationship. The key ingredient to this observation is quantum reference frames (QRFs), which constitute a universal tool for dealing with symmetries in quantum systems. They define a split between redundant and physical information in gauge systems, thereby establishing a notion of encoding in that context. This leads to an exact dictionary between (group-based) quantum error correcting codes and QRF setups. In stabilizer codes, this uncovers a correspondence between errors and QRFs: every maximal set of correctable errors generates a unique QRF, and each QRF is associated with a unique class of correctable errors. This allows for a reinterpretation of the Knill-Laflamme condition and novel insights into the relation between correctability and redundancy. The dictionary also reveals a novel error duality, based on Pontryagin duality, and somewhat akin to electromagnetic duality. Time permitting, I will illustrate these findings in surface codes, which can be understood as both codes and lattice gauge theories. These findings may find use in code design and quantum simulations of gauge theories.