Simple example for the sub-optimality of MWPM vs Maximum likelihood decoder

Question

I understand that the Minimum Weight Perfect Matching (MWPM) decoder tries to identify the single most probable error configuration, while a maximum likelihood decoder aims to find the most probable coset of errors (those equivalent up to a stabilizer).

I’m looking for a straightforward example on a surface code where MWPM produces a solution that does not lie in the most probable coset. I found some examples with non-uniform error probabilities, but since the surface code threshold is different between the two decoders for uniform error probabilities, I guess there should be an example without using non-uniform error probabilities.

Does anyone have a concrete example or a reference illustrating a case where MWPM selects an error not contained in the most probable coset?

Answer 1

Consider the violated star stabilizers indicated by the two yellow circles in the image below. The correction indicated with the blue line has weight 12 and the correction indicated with the red line has weight 13.

The blue line is the only correction of length 12, while there are topologically equivalent red lines that have length 13:

All corrections of weight 13 are indicated by the red grid:

A maximum likelihood decoder will apply the correction indicated by the red line if the probability of the errors indicated by the red lines is higher than the probability of the error indicated by the blue line. The minimum weight decoder will apply the correction indicated by the blue line.

For this specific example and toy noise model, this happens if the probability of a single error is around 30%, as shown in the plot below.

These screen shots are from Christopher Chubb's lecture at the IBM summer school.