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I have heard multiple times the claim that three rounds of syndrome measurement required for fault-tolerance in the surface code. I'm not sure what situation would require this, as I think less would suffice. Unfortunately, all these mentions were made in cases that I can't cite for you (like an off-hand comment), and I never followed up. But I wondered if anyone here might know what situation this is referring to.

In my claims that less would suffice, I am specifically imagining a surface code with standard $\sigma^z$ and $\sigma^x$ type stabilizers, and for which the logical $Z$ is composed of all $\sigma^z$s. I'm also assuming that it starts its life placing all physical qubits in the $|0\rangle$ state. This will result in the logical $|0\rangle$ state once stabilizer measurements are performed. For a logical $|0\rangle$, a string of physical $|1\rangle$s is placed across the code. Measurement of the logical $Z$ can be done at readout time by measuring all physical qubits in the $\sigma^z$s basis. This provides information that can infer the final $Z$ value, as well as a final state for the $\sigma^x$ stabilizers that can be used to correct it.

If we add in a single ancilla-assisted syndrome round between initialization and readout, we have something a bit more quantum. In the worst case, we could just ignore the results of these measurements. Then they just become an extra source of noise in the case described above. This extra noise wouldn't be enough to effect the fault-tolerance, even for distance 3. The case where we use the measurement results to decode should even be an improvement. And so on for the addition of more rounds.

From this, I see nothing special about the case of three rounds with regards to fault-tolerance. So I guess these claims referred to a slightly different situation.

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