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I am trying to understand fault tolerance for gate gadgets a bit better. Here Gottesman provides the following definition: enter image description here

I generally understand this definition but I have two questions.

(1) Why is it the case that we don't assume the input is error free? Can't we just do error correction on the data before inputting it into a gate gadget? In other words, why is the $r_i$-filter necessary?

(2) Usually gate gadgets involve ancillas. Do we usually assume the ancilla state is input into the gadget error free or not? If there was an error on an ancilla input is that considered part of one of the $s$ faults of the gadget or is it just an input error (part of one of the $r_i$ blocks)?

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Why is it the case that we don't assume the input is error free? Can't we just do error correction on the data before inputting it into a gate gadget?

Errors can happen at the very end of an error correction cycle, too late in the circuit for the errors to propagate into the measurements of that cycle. You can't correct unmeasured errors, so those errors cannot possibly be corrected before the logical gate starts.

Also, unless you're using a one shot code, you need multiple more rounds of information before a recent error can be resolved and corrected. So it's not just errors from the very end of the last error correction cycle, it's errors from the last few cycles that are waiting to be decoded.

Do we usually assume the ancilla state is input into the gadget error free or not?

Ancilla states can have errors. If we had perfect ancilla qubits, what would we need error correction for? We'd just use the perfect qubits for everything, instead of only as helper qubits to measure stabilizers.

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