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Quantum stabilizer codes often struggle with syndrome errors due tomeasurement imperfections. Typically, multiple rounds of syndrome extractionare employed to ensure reliable error information. In this paper, we considerphenomenological decoding problems, where data qubit errors may occur betweenextractions, and each measurement can be faulty. We introduce generalizedquantum data-syndrome codes along with a generalized check matrix thatintegrates both quaternary and binary alphabets to represent diverse errorsources. This results in a Tanner graph with mixed variable nodes, enabling thedesign of belief propagation (BP) decoding algorithms that effectively handlephenomenological errors. Importantly, our BP decoders are applicable to generalsparse quantum codes. Through simulations, we achieve an error threshold ofmore than 3\% for quantum memory protected by rotated toric codes, using solelyBP without post-processing. Our results indicate that $d$ rounds of syndromeextraction are sufficient for a toric code of distance $d$. We observe that athigh error rates, fewer rounds of syndrome extraction tend to perform better,while more rounds improve performance at lower error rates. Additionally, wepropose a method to construct effective redundant stabilizer checks forsingle-shot error correction. Our simulations show that BP decoding remainshighly effective even with a high syndrome error rate.

Generalized quantum data-syndrome codes and belief propagation decoding for phenomenological noise