  Quantum error correction is necessary to protect logical quantum states and
operations. However, no meaningful data protection can be made when the
syndrome extraction is erroneous due to faulty measurement gates. Quantum
data-syndrome (DS) codes are designed to protect the data qubits and syndrome
bits concurrently. In this paper, we propose an efficient decoding algorithm
for quantum DS codes with sparse check matrices. Based on a refined belief
propagation (BP) decoding for stabilizer codes, we propose a DS-BP algorithm to
handle the quaternary quantum data errors and binary syndrome bit errors.
Moreover, a sparse quantum code may inherently be able to handle minor syndrome
errors so that fewer redundant syndrome measurements are necessary. We
demonstrate this with simulations on a quantum hypergraph-product code.
