about noisy topological CSS states where thermal phase transition in a ferromagnetic spin model is mapped to a critical stability against bit-flip noise in the corresponding topological CSS code[@zare18]. In particular, it has been shown that phase transition in two-dimensional (2D) Ising model is mapped to a coherence-noncoherence phase transition in the 2D Toric code model under a bit-flip noise [@zareimon].

In addition to the dual correspondence, connection between thermal phase transition in classical spin models and noisy topological codes has also been considered in a different framework [@Dennis2002; @Katzgraber2009; @q1; @q2; @q3; @q4; @3dcolor], where the error threshold of noisy topological codes is mapped to the transition point of the random bond Ising models on the Nishimori line [@nishi] . Accordingly, it seems that such problem should be related to the dual correspondence between classical spin models and quantum CSS states. If it is correct, it means that the same dual correspondence also relates the error threshold of a topological CSS state defined on a hypergraph $H$ with the phase transition point of a random bond Ising system defined on $\tilde{H}$.

In this paper, we start from the dual correspondence for the 2D random bond Ising model where the 