  It is known that noisy topological quantum codes are related to random bond
Ising models where the order-disorder phase transition in the classical model
is mapped to the error threshold of the corresponding topological code. On the
other hand, there is a dual mapping between classical spin models and quantum
Calderbank-Shor-Stean (CSS) states where the partition function of a classical
model defined on a hypergraph $H$ is written as an inner product of a product
state and a CSS state on dual hypergraph $\tilde{H}$. It is then interesting to
see what is the interpretation of the classical phase transition in the random
bond Ising model within the framework of the above duality and whether such an
interpretation has any connection to the error threshold of the corresponding
topological CSS code. In this paper, we consider the above duality relation
specifically for a two-dimensional random bond Ising model. We show that the
order parameter of this classical system is mapped to a coherence order
parameter in a noisy Toric code model. In particular, a quantum phase
transition from a coherent phase to a non-coherent phase occurs when the
initial coherent state is affected by two sequences of bit-flip quantum
channels where a quenched disorder is induced by measurement of the errors
after the first channel. On the other hand, the above transition is directly
related to error threshold of the Toric code model. Accordingly, and since the
noisy process can be applied to other topological CSS states, we conclude that
the dual correspondence can also provide a useful tool for the study of error
thresholds in different topological CSS codes.
