  Quantum stabilizer codes often face the challenge of syndrome errors due to
error-prone measurements. To address this issue, multiple rounds of syndrome
extraction are typically employed to obtain reliable error syndromes. In this
paper, we consider phenomenological decoding problems, where data qubit errors
may occur between two syndrome extractions, and each syndrome measurement can
be faulty. To handle these diverse error sources, we define a generalized check
matrix over mixed quaternary and binary alphabets to characterize their error
syndromes. This generalized check matrix leads to the creation of a Tanner
graph comprising quaternary and binary variable nodes, which facilitates the
development of belief propagation (BP) decoding algorithms to tackle
phenomenological errors. Importantly, our BP decoders are applicable to general
sparse quantum codes. Through simulations of quantum memory protected by
rotated toric codes, we demonstrates an error threshold of 3.3% in the
phenomenological noise model. Additionally, we propose a method to construct
effective redundant stabilizer checks for single-shot error correction.
Simulations show that BP decoding performs exceptionally well, even when the
syndrome error rate greatly exceeds the data error rate.
