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Are there any quantum error-correction codes which have been found to exhibit high thresholds for either Pauli-based depolarizing noise or Erasure noise channels, but not for both? Are there reasons to suspect that an error-correction code would perform better or worse under either of these channels?

A naive example could perhaps be the repetition code family, which can correct for erasure, but not general Pauli-based depolarizing noise.

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  • $\begingroup$ Please define better your question. In depolarizing noise, do you mean unheralded noise, vs the heralded erasure? And when do you say, "high thresholds for either...", high with respect to what? In general, any code will perform better against heralded noise than again unheralded non-biased noise. $\endgroup$ Commented Apr 27, 2023 at 18:17
  • $\begingroup$ heralded noise, and "high" in the general sense. For example see these papers: journals.aps.org/prresearch/pdf/10.1103/… and arxiv.org/abs/1611.04256. Here the authors report 25-50% thresholds under their erasure noise channel. $\endgroup$ Commented Apr 27, 2023 at 18:22

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Erasure is heralded. Therefore, in the absence of other noise channels you can always correct against it up to the percolation threshold (24.9% for 3D cubic lattice, 50% for 2D squared lattice). However, using different methods of redundant encoding or post-selection, you can reach much higher rates. so in general, the more redundant the lattice, the better it preforms against erasure (but the worse it performs against unheralded noise). For example, see https://arxiv.org/abs/2212.04834, and many others.

for depolarizing noise in your gates and measurements, you need a 3D lattice. If your noise is symmetric (not biased for X, Z or Y noise), so a symmetric lattice (namely a lattice with the same primal and dual syndrome graphs) will perform best. In general, the least edges connected to a node (syndrome) in the syndrome graphs, and the least gates performed on a qubit in the physical lattice, the better the code will perform. Although naively these two requirements are opposite, see in https://arxiv.org/abs/1810.09621 that it is not always the case.

Also note that "a code able to correct t unlocated errors is also able to correct 2t located errors" (https://arxiv.org/abs/quant-ph/0601066)

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