  A promising strategy to protect quantum information from noise-induced errors
is to encode it into the low-energy states of a topological quantum memory
device. However, readout errors from such memory under realistic settings is
less understood. We study the problem of decoding quantum information encoded
in the groundspaces of topological stabilizer Hamiltonians in the presence of
generic perturbations, such as quenched disorder. We first prove that the
standard stabilizer-based error correction and decoding schemes work adequately
well in such perturbed quantum codes by showing that the decoding error
diminishes exponentially in the distance of the underlying unperturbed code. We
then prove that Quantum Neural Network (QNN) decoders provide an almost
quadratic improvement on the readout error. Thus, we demonstrate provable
advantage of using QNNs for decoding realistic quantum error-correcting codes,
and our result enables the exploration of a wider range of non-stabilizer codes
in the near-term laboratory settings.
