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The toric code and other popular codes can be decoded using minimum weight perfect matching. Is this an optimal decoder? Here by optimal, I mean it gives the best logical error rate vs physical error rate performance in depolarizing channel. "Threshold" is often used to characterize toric codes, does that assume optimal decoding or a particular type?

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Is [minimum weight perfect matching] an optimal decoder?

No, it's not optimal. For example, it uses the weight of the shortest path between two detection events as an approximation for the contributions of all topologically equivalent paths. An optimal decoder would exactly compute the contributions of all possible errors consistent with the symptoms, instead of focusing on the most likely one.

"Threshold" is often used to characterize toric codes, does that assume optimal decoding or a particular type?

Thresholds are always relative to a decoder. And ideally that decoder is something that runs in polynomial time, instead of a hypothetical optimal decoder which likely takes exponential time.

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  • $\begingroup$ It makes perfect sense that performance and threshold are decoder dependent. Unfortunately I see a lot of papers refer to "the threshold of the code" without reference to the decoding algorithm. $\endgroup$
    – unknown
    Oct 18 at 18:30
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    $\begingroup$ @unknown Oh it's even worse than that. They all refer to different error models too. Sometimes it's just single qubit depolarizing noise between rounds, sometimes there's no measurement errors sometimes there isn't, sometimes it's full circuit noise, sometimes a single qubit depolarization error that triggers picks randomly between {I,X,Y,Z} and sometimes it picks between {X,Y,Z}. You can find papers saying the surface code threshold is ~1% and other papers saying it's ~10% and neither is "wrong". $\endgroup$ Oct 18 at 18:42

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