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. We determine 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. In either case, there exist codes which approximately achieve the scaling of our bound and become good covariant error-correcting codes. We show this using three examples: an infinite-dimensional rotor extension of the three-qutrit secret-sharing code, an infinite-dimensional five-rotor perfect code, and a many-body Dicke-state code. Furthermore, we prove an approximate version of the Eastin-Knill theorem that puts a severe quantitative limit on a code’s ability to correct erasure errors if it admits a universal set of transversal logical gates. This bound goes to zero only inversely in the logarithm of the local physical subsystem dimension. We provide two examples of codes circumventing this bound: a many-body generalized W-state code and families of codes whose transversal gates form a general group G.