# What does "perfect measurement" mean in the context of fault tolerance?

I am reading the paper Quantum error correction with only two extra qubits.

They say there that (in the section on the [[5, 1, 3]] code):

If a flag is raised or a syndrome is nontrivial, then the subsequent unflagged syndrome extractions are perfect, and suffice to correct either a possibly correlated error (if the flag is raised) or a weight ≤ 1 error (if no flag is raised but the syndrome is nontrivial)

What does "perfect extraction" mean in this context? Surely the unflagged circuits are just as susceptible to errors as the flagged ones, no? I hope it's not just a statement that says that since the distance is 1, and a single error was already raised, then we assume a-priori that the technique can only succeed if the unflagged round has exactly zero errors?

what does "perfect extraction" mean in this context?

In the parent bullet point, they say "if [...] there is at most one faulty gate". So they're just saying if you used your faulty gate to break the flag, then you can't also use it to break the measurement.

The reason you would start by focusing on what happens with single errors is because all other errors are at least $$p$$ times less important, where $$p$$ is the physical error rate. If adding the flag qubits raises the minimum cost of a logical failure from 1 physical error to 2 physical errors, that's progress!

Of course you then consider two errors, and three, and four, and etc. But just showing "what used to be distance $$d$$ is now distance $$2d$$" is a pretty good start.

• wait, hold on... are you saying this corrects any weight 2 error? I thought it can only distinguish weight 1 errors + the set IIIII, IIZXI, IXZXI, IY ZXI, IZZXI, IIIXI, IIXXI, IIYXI no? also, just to make sure I understand: "perfect extraction" in this context means "must occur without faults for the error correction procedure to be successful"? Also, when he says "error correction procedure", he means the entire error correction round from start to finish, right?
– Lior
Feb 16, 2022 at 20:06
• @Lior No I'm not saying that. First, it might only detect instead of correct. Second, they might be focusing on detecting specific problematic errors like hook errors. Feb 16, 2022 at 20:11
• exactly - that's what they're focusing on. You can see that in fig. 2. Though he doesn't say that explicitly for some reason. BTW, have you given any though to the other questions? Help me Craig, you're my only hope :-)
– Lior
Feb 16, 2022 at 20:17
• @Lior Typically you're supposed to ask one question per question. It makes it too confusing to answer. Feb 16, 2022 at 20:23
• got it, will do
– Lior
Feb 16, 2022 at 20:28