  Quantum error correction codes (QECC) are a key component for realizing the
potential of quantum computing. QECC, as its classical counterpart (ECC),
enables the reduction of error rates, by distributing quantum logical
information across redundant physical qubits, such that errors can be detected
and corrected. In this work, we efficiently train novel {\emph{end-to-end}}
deep quantum error decoders. We resolve the quantum measurement collapse by
augmenting syndrome decoding to predict an initial estimate of the system
noise, which is then refined iteratively through a deep neural network. The
logical error rates calculated over finite fields are directly optimized via a
differentiable objective, enabling efficient decoding under the constraints
imposed by the code. Finally, our architecture is extended to support faulty
syndrome measurement, by efficient decoding of repeated syndrome sampling. The
proposed method demonstrates the power of neural decoders for QECC by achieving
state-of-the-art accuracy, outperforming {for small distance topological
codes,} the existing {end-to-end }neural and classical decoders, which are
often computationally prohibitive.
