Linear Programming Bounds for Approximate Quantum Error Correction Over Arbitrary Quantum Channels

While quantum weight enumerators establish some of the best upper bounds on minimum distance error-correcting codes, these are not optimized to quantify performance codes under effect arbitrary channels that describe bespoke noise models. Herein, for any Kraus decomposition given channel, we introduce corresponding naturally generalize Shor-Laflamme enumerators. We an indirect linear relationship between generalized by introducing auxiliary exact enumerator completely quantifies code’s projector, and is independent underlying process. By additionally working within framework approximate error correction, a general constructing program infeasible whenever correcting with parameters do exist. Our programming allows us non-existence certain approximately correct amplitude damping errors, obtain non-trivial maximum dimension broad family permutation-invariant codes.