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In the paper Surface code quantum computing by lattice surgery, for rough merging, they state that measuring the $X$ stabilisers at the boundary is equivalent to $X_{L}X_{L}$.

What I don't understand is what the new $X$ stabiliser measurements spanning the old boundary are meant to be. and how they're product gives these logical operators.

Do they mean that, using the 3 pink data qubits, they construct 3 new $X$ stabilizer measurements, and then the product of those 3 stabilizers would be equivalent to 2 logical $X$ operators running parallel to the old boundary? If this is the case, then why would the rounds of error correction we do on both the 3 qubits and 2 surfaces allow us to somehow arrive at this

If they are taking the existing $X$ stabilisers, then I don't see how the product of them even givens you the product of the 2 logical $X$ operators of each surface, given they act on different qubits, unless the idea is that the data qubits in the boundaries of the lattice disappear.

image showing lattice merging from paper

GaussStrife
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The $X$ stabilizers to be measured are the ones indicated. You have two three-qubit stabilizers at the top and bottom and two four-qubit stabilizers. enter image description here

Note that each pink qubit is acted on by two stabilizer measurements. Hence that the product of all these stabilizers is equal to two columns of $X$ measurements, one on the rightmost column of the left patch and one on the leftmost column of the right patch. This is the measurement $X_LX_L$.

Navneeth Ramakrishnan
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