Continuous symmetries and approximate quantum error correction

Presenting Author: Sepehr Nezami, Stanford University
Contributing Author(s): Philippe Faist, Victor V. Albert, Grant Salton, Fernando Pastawski, Patrick Hayden, John Preskill

Quantum error correction and symmetries are relevant to many areas of physics, including many-body systems, holographic quantum gravity, and reference-frame error-correction. Here, 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, we provide examples of codes that approximately achieve the scaling of our bound: 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 examples of codes circumventing the Eastin-Knill theorem: random unitary covariant codes, many-body generalized W-state code, and families of codes whose transversal gates form a general group G. 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.

(Session 9b : Monday from 5:45pm - 6:15pm)


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