I am interested in qubits which are biased in noise. For my purpose I assume that I have only bit-flip (i.e. Pauli $X$) noise in my algorithm.
In the presence of such noise, quantum gates can convert this bit-flip into phase flip (Pauli $Z$). For instance $HX=ZH$: a Hadamard will convert an $X$ error to a $Z$ error.
I am in particular interested in cNOT gates. By construction, they will satisfy:
$$cNOT X_1 = X_1 X_2 cNOT$$ $$cNOT X_2 = X_2 cNOT$$
Because of that we could believe that a cNOT does the job. However, problems come when we consider the continuous evolution for the gate. For instance, a way to implement a $cNOT$ is through the Hamiltonian:
$$H=\hbar g |1\rangle \langle 1| |-\rangle \langle - |$$
Because this Hamiltonian involves Pauli $Z$, if an $X$ error occurs during the continuous evolution it might be converted to a $Z$ error.
I am aware of this paper which ensure that cNOT preserve the $Z$ noise during the continuous evolution. However I would like to preserve the $X$ noise.
Is there any paper which shows that it is possible to create a cNOT that preserves bit-flip noise?