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I'm reading Horsman et al. "Surface code quantum computing by lattice surgery" and I'm wondering about the rotated surface code.

Consider Figure 13:

enter image description here

This is supposed to have distance 5. But in (c), an $X$ error on the top left qubit and the qubit beneath it would be undetectable (brown plaquettes are $X$-stabilizer measurements). If there were a stabilizer measurement on every outside edge, this problem wouldn't occur. What am I misunderstanding?

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an X error on the top left qubit and the qubit beneath it would be undetectable

That error would be detected by the flipping of the four body Z stabilizer adjacent to the lower qubit you operated on:

enter image description here

If you or I got X and Z mixed up, then the error is undetectable, but it corresponds to a topologically trivial cycle from a boundary to itself, so it has no effect on the logical qubit:

enter image description here

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The 𝑋 error on the top left qubit and the qubit beneath it indeed produce the same measurement syndrome, but this does not mean they are uncorrectable. Lets call 𝑋 on the top left qubit $𝑋_{topleft}$ and 𝑋 beneath it $X_{beneath}$. $X_{topleft} 𝑋_{beneath}$ is a stabilizer and acts as the identity gate on the logical qubit. So we can choose to correct both errors with either $𝑋_{topleft}$ of $X_{beneath}$.

Take a look at: How does Surface-17 tell apart Z errors on Db and Dc?

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  • $\begingroup$ How can I see that $X_{topleft}X_{beneath}$ is a stabilizer? $\endgroup$
    – Sam Jaques
    May 6 '20 at 17:12
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I guess you are confusing Stabilizer types.

Brown plaquettes(X Stabilizer) & Yellow plaquettes(Z Stabilizer)

So if there is Error Z_topleft or Error Z_beneath, it gives same Syndrome. if this is your question,

Imagine if there are only Z Error occurred on the Rotated Surface Code ( Figure c).

Z_topleft Z_beneath is one of stabilizer. So we can choose Z_topleft or Z_beneath as a correction operator

  1. Z_topleft Error
  • Correction Operator (Z_topleft)

    Z_topleft*Z_topleft = I

  • Correction Operator (Z_beneath)

    Z_topleft*Z_beneath -> Stabilizer

  1. Z_beneath
  • Correction Operator (Z_topleft)

    Z_beneath*Z_topleft -> Stabilizer

  • Correction Operator (Z_beneath)

    Z_beneath*Z_beneath = I

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