Performance of surface codes with imperfect erasure detection

It is known that surface codes are highly effective at correcting erasures, that is, unknown errors at known locations. With this motivation, one recent work by Wu et al. [Nat Commun 13, 4657 (2022)] tailored the noise channel of a neutral atom qubit so that the dominant decay errors in the system could be detected with η=100% efficiency. The noise channel in this system is described by erasure errors which occur at high rate and Pauli errors which occur at a low rate. In this work, we consider the situation when the erasure-conversion efficiency (η) is not 100%. We numerically analyze the performance of the surface code under this noise model, estimate the thresholds, and determine logical error scaling as a function of η. Our results are important to design near-term experiments and also identify other platforms where it may be possible to achieve good (albeit imperfect) erasure conversion.