# Is there benefit to extra stabilizers in a rotated surface code?

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:

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?

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:

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:

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 $$X_{\text{topleft}}$$ and $$X$$ beneath it $$X_{\text{beneath}}$$. Then, $$X_{\text{topleft}} X_{\text{beneath}}$$ is a stabilizer and acts as the identity gate on the logical qubit.

In the case of a single error $$X_{\text{topleft}}$$ (or $$X_{\text{beneath}}$$) we can choose to correct both errors with either $$X_{\text{topleft}}$$ or $$X_{\text{beneath}}$$. Both operations $$X_{\text{either}}X_{\text{topleft}}$$ and $$X_{\text{either}}X_{\text{beneath}}$$ being logical identities.

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

• How can I see that $X_{topleft}X_{beneath}$ is a stabilizer? Commented May 6, 2020 at 17:12

I guess you are confusing Stabilizer types.

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

So if there is Error $$Z_{\text{topleft}}$$ or Error $$Z_{\text{beneath}}$$, it gives the same Syndrome. If this is your question,

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

$$Z_{\text{topleft}}$$ $$Z_{\text{beneath}}$$ is one of stabilizer. So we can choose $$Z_{\text{topleft}}$$ or $$Z_{\text{beneath}}$$ as a correction operator

1. $$Z_{\text{topleft}}$$ Error
• Correction Operator ($$Z_{\text{topleft}}$$)

$$Z_{\text{topleft}}Z_{\text{topleft}}$$ = $$I$$

• Correction Operator ($$Z_{\text{beneath}}$$)

$$Z_{\text{topleft}}Z_{\text{beneath}}$$ -> Stabilizer

1. $$Z_{\text{beneath}}$$ Error
• Correction Operator ($$Z_{\text{topleft}}$$)

$$Z_{\text{beneath}}Z_{\text{topleft}}$$ -> Stabilizer

• Correction Operator ($$Z_{\text{beneath}}$$)

$$Z_{\text{beneath}}Z_{\text{beneath}}$$ = $$I$$