  Topological quantum codes are favored because they allow qubit layouts that
are suitable for practical implementation. An $N$-qubit topological code can be
decoded by minimum-weight perfect matching (MWPM) with complexity
$O(\text{poly}(N))$ if it is of CSS-type. Recently it is shown that various
quantum codes, including non-CSS codes, can be decoded by an adapted belief
propagation with memory effects (denoted MBP) with complexity almost linear in
$N$. In this paper, we show that various two-dimensional topological codes, CSS
or non-CSS, regardless of the layout, can be decoded by MBP, including color
codes and twisted XZZX codes. We will comprehensively compare these codes in
terms of code efficiency and decoding performance, assuming perfect error
syndromes.
