Continuous symmetries and approximate quantum error correction

Quantum error correction and symmetries are relevant to many areas of physics, including many- body quantum systems, holographic quantum gravity, and reference-frame error-correction [Hayden et al., arXiv:1709.04471]. Here, we show that any code is fundamentally limited in its ability to approximately error-correct against erasures at known locations if it is covariant with respect to a continuous local symmetry. Our bound vanishes either in the limit of large individual subsystems, or in the limit of a large number of subsystems, and is approximately tight in these regimes. Furthermore, we prove an approximate version of the Eastin-Knill theorem that quantifies a code’s ability to correct erasure errors if it admits a universal set of transversal logical gates. The bound is in terms of the local physical subsystem dimension. We provide a collection of example codes illustrating our bounds in different regimes. In the context of the AdS/CFT correspondence, our approach provides insight into how time evolution in the bulk corresponds to time evolution on the boundary without violating the Eastin-Knill theorem, and our five-rotor code can be stacked to form a covariant holographic code.