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I'm looking into this paper from DiCarlo's group Scalable quantum circuit and control for a superconducting surface code. I don't understand how it's supposed to identify specific single-qubit errors, specifically how it tells apart single qubit X or Z errors on the data qubits belonging only to one of the weight-4 syndromes.
Does it? Maybe that's not even required, if it really isn't, why?
I have added the layout from the paper.
A Z error on Db will fire Xb and Xa. A Z error on Dc will fire Xa. Thus these two are distinguishable.
If a X error occurs on Dc this will fire Zb. This can be corrected by applying X on Dc.
If a X error occurs on Db this will also fire Zb. It is also corrected by applying X on Dc. At the end Db and Dc have been flipped, this does not change the logical state, because X-Db X-Dc is one of the stabilizers. This stabilizer is measured by Xa. Remember that applying a stabilizer is the same as applying the identity gate.
Distinguishing $X$ and $Z$ errors is easy. $X$ errors anti-commute with the $Z$-type stabilizers, and so when you perform a measurement of those parity checks, you get and answer '1'. Similarly, $Z$ errors give you a '1' answer only on the $X$-type parity checks.
Also, note that, in the bulk (i.e. not on the edges), you never get a '1' on only one weight-4 parity check syndrome. They always come in pairs. You can never identify exactly what error happened (although you might make a good guess) but that doesn't prevent you from correcting the error. That's the whole point of a topological system. If a single $X$ error occurred somewhere, you don't have to apply the same $X$ to correct for it. Any sequence of $X$ operations that creates a closed loop on the lattice (without going off the edge) will also correct the error (and could equally have been an error sequence that gave the same syndrome).
