## How does Surface-17 tell apart Z errors on Db and Dc?

1

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?

0

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.

1

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).

That's how it's supposed to work for a infinite surface code, but surface-17 is just a 3x3 grid of data qubits with 4 weight-4 syndromes and 4 weight-2 syndroms – Ilya Besedin – 2020-01-09T11:50:34.940

The surface code is never infinite. The whole point is that it has a finite boundary. And the same understanding is supposed to work. – DaftWullie – 2020-01-09T11:52:45.780