Approximate quantum codes for amplitude damping from variational quantum learning

We introduce a machine learning approach to discover new approximate quantum error-correcting (AQEC) codes. Specifically, we use variational quantum circuits to parameterize the encoding channels and use the violation of the Knill-Laflamme (KL) conditions directly as the loss function. This protocol's advantage lies in its ability to incorporate arbitrary noise channels, enabling us to find AQEC codes adaptive to specific noise models. In this work, we demonstrate new codes found through this protocol for two types of amplitude damping noise channels: 1) everywhere noise; 2) noise at unknown positions. For everywhere noise, we found a ((4, 1)) amplitude damping code that can approximately correct a single amplitude damping error, with better AQEC capability compared to the previously known code. With noise at unknown position, we were able to find codes with parameter ((n, n-1)) for arbitrary n, capable of approximately correcting one amplitude damping error occurring at an unknown position.