  Finding efficient decoders for quantum error correcting codes adapted to
realistic experimental noise in fault-tolerant devices represents a significant
challenge. In this paper we introduce several decoding algorithms complemented
by deep neural decoders and apply them to analyze several fault-tolerant error
correction protocols such as the surface code as well as Steane and Knill error
correction. Our methods require no knowledge of the underlying noise model
afflicting the quantum device making them appealing for real-world experiments.
Our analysis is based on a full circuit-level noise model. It considers both
distance-three and five codes, and is performed near the codes pseudo-threshold
regime. Training deep neural decoders in low noise rate regimes appears to be a
challenging machine learning endeavour. We provide a detailed description of
our neural network architectures and training methodology. We then discuss both
the advantages and limitations of deep neural decoders. Lastly, we provide a
rigorous analysis of the decoding runtime of trained deep neural decoders and
compare our methods with anticipated gate times in future quantum devices.
Given the broad applications of our decoding schemes, we believe that the
methods presented in this paper could have practical applications for near term
fault-tolerant experiments.
