# MWPM algorithm for depolarizing-like noise channel (i.e. non-zero probability of $Y$ errors)

I would like to understand how the MWPM decoder can be used in the case of a noise model where $$X$$,$$Y$$ and $$Z$$ can all be introduced with a comparable probability (one example would be the depolarizing noise model).

My current understanding of the decoder relies upon considering two graphs when the error occur: one associated to $$Z$$ errors and one associated to $$X$$ errors, and to decode the graph independently by applying a min-weight correction $$R_X$$ that fixes the $$X$$ errors, and then a min-weight correction $$R_Z$$ that fixes the $$Z$$ errors ($$X$$ and $$Z$$ are dealt as independent events). When I mean "fix", I mean that the net operation Error+Correction is equal to some stabilizer.

Because we treat $$X$$ and $$Z$$ errors independently, the "logic" of the MWPM in this case is based upon the following noise model:

$$\mathcal{N}(\rho)=(1-(p_X+p_Z)) \rho + p_X X \rho X + p_Z Z \rho Z$$

As soon as there is a probability to have $$Y$$ errors, with a probability $$p_Y$$ both an $$X$$ and $$Z$$ error will be introduced. How is the decoder modified in such a case?

I guess (but I'm unsure, see below) that keeping the same strategy of independent decoding would still do the job. With respect to the graph associated to $$X$$ errors, everything behave as if $$p_X \to p_X + p_Y$$, and with respect to the graph associated to $$Z$$ errors, everything behave as if $$p_Z \to p_Z + p_Y$$.

However, maybe one can optimize the MWPM decoder to acknowledge the correlations between $$X$$ and $$Z$$ (and maybe it would lead to better performances?). In this case I don't see how the decoding graphs are constructed?

In summary, my questions are:

• Do you agree that if $$Y$$ error are introduced (with a probability to occur that is comparable to $$X$$ and $$Z$$), the decoder that decodes independently $$X$$ and $$Z$$ would still work efficiently? Or maybe because it ignores correlations it completely ruins the protection (because "for some reason" events of errors less than $$t$$, for a code distance $$d=2t+1$$, would now lead to logical errors).
• If there is a way to "optimize" the MWPM to deal with cases where $$Y$$ error are also introduced (for instance depolarizing noise), I would be interested by a nice pedagogic reference on the matter.

As a last comment if it can guide the answer: I am not looking for the "supper dubber state of the art method". My goal is to get better intuition in order to make a small simulation for my own pedagogic purposes.

First of all, I think that the protection would not be "ruined". I mean this because even if the $$X$$ and $$Z$$ graphs are independently decoded, whenever a qubit is determined to have suffer from an error in both of the graphs, the resulting error would be a $$Y$$ error. Therefore, such approach would also correct some errors of such nature. However, since the probability of occurrence for $$Y$$ errors in a depolarizing channel is the same as for $$X$$ and $$Y$$ errors, the performance of the decoder will be lowered substantially due to the fact that considering $$X$$ and $$Z$$ errors in a independent manner implies that the probability of a $$Y$$ to occur would be $$p_y=p_xp_z$$, which is lower than what it is expected. Thus, the performance loss will be a function of the correlation as you were discussing, the optimal performance of such naive decoder will be for the named independent $$XZ$$ channel, where $$p_y=p_xp_z$$ is fulfilled.