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One of the key features of quantum error correction that makes it different from classical error correction. When code is non-degenerate, an error $E$ takes codewords to different words. codewords.

Formally, for all basis vectors $c_i$, we get $\langle c_i, E c_i \rangle=0$.

This behavior is much more closer to classical error correction. Does this make decoding(/detecting) of non-degenerate codes easier since there is a very well developed classical machinery in many cases?

The traditionally way to detect or correct errors (at least for stabilizer codes) is getting syndrome by looking at commuting relations and then do syndrome decoding. The second step of syndrome decoding resembles closely to classical counterpart. Does that make degenerate error intrinsically harder to detect, or to put it other way, easier for non-degenerate errors?

The motivation for this question is trying to understand can there be a reason where one prefers non-degenerate codes over degenerate codes.

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As you say, non-degenerate codes have a lot of well-understood machinery that's brought in from the classical side. That helps us from a conceptual stance, and a mathematical one (making rigorous results easier to prove), although, in terms of practical implementation, doesn't necessarily mean that one is easier to implement than the other.

Errors in a degenerate code do not need to be any harder to detect. For example, if you look at the Toric Code, this is degenerate but it's also a stabilizer code, so experimentally, it's just the same. If anything, it's easier because all the observables that we need to measure are local, each comprising just four neighbouring qubits.

Why might you prefer non-degenerate models? I think, really, it comes down to the rigour of the mathematics behind some of the results. For example, things like the quantum Hamming bound apply to non-degenerate codes. We know where we stand. Actually, that gives degenerate codes a potential advantage in that they might be better than the non-degenerate ones. It's just hard to prove. I gave some numerical indications of this at some point for the Toric code: https://arxiv.org/abs/1208.4924. I think I also linked to some previous results for other degenerate codes.

To give an indication of the challenge with the Toric code: an $N\times N$ lattice has a code distance $O(N)$, while there exist non-degenerate codes of the same size with distance $O(N^2)$. So, on paper, this looks bad for degenerate codes. Nevertheless, one can prove (with some effort) that if the error model is independent per-qubit errors, you can correct for an error rate that is finite, i.e. almost all combinations of $O(N^2)$ errors (up to some threshold value) that might typically arise can be corrected. So, we know some of these statements for the Toric code, but let's say you've come up with some wonderful knew degenerate code. How does it behave? No clue! But if you come up with a wonderful new non-degenerate code, you can immediately know a lot of its properties.

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  • $\begingroup$ I understand that practical implementations could be even easier for degenerate codes. Let me rephrase this small question in particular: Will classical machinery syndrome decoding work equally for degenerate or non-degenerate codes? $\endgroup$ – Root Jun 1 at 14:51
  • $\begingroup$ I don't believe so. Degenerate codes, if you're working in this regime where there are more errors than the distance of the code, require some classical processing that (roughly) takes a global overview of what errors have happened, and has to apply some sort of heuristic to guess what the correction should be. There's been a lot of work on doing this well, but I don't think you'd class it under the classical machinery for syndrome decoding. $\endgroup$ – DaftWullie Jun 1 at 15:52
  • $\begingroup$ See, for example, arxiv.org/abs/0911.0581 $\endgroup$ – DaftWullie Jun 1 at 15:55
  • $\begingroup$ So will it be right to say that non-degenerate codes can directly use classical syndrome decoding (for example CWS codes) but degenerate codes (at least in the regime of degenerate errors) lack this feature since they are looking to correct larger errors. $\endgroup$ – Root Jun 1 at 17:23
  • $\begingroup$ I don't know that this is true of all non-degenerate codes. CSS codes are carefully constructed out of classical codes so that syndrome extraction closely parallels the classical case. However that doesn't mean there aren't intrinsically quantum non-degenerate codes which don't have a classical equivalent. $\endgroup$ – DaftWullie Jun 2 at 6:51

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