Realization of Surface code, can an eigenvalue of -1 be taken?

Question

I have a question about the implementation of Surface Code. I understand that Surface Code is a stabilizer state defined by plaquette operators and vertex operators, meaning it is a state where the eigenvalues of the plaquette and vertex operators are 1.

However, according to this documentSurface code: Towards practical large-scale quantum computation, the indirect measurement of operators is implemented as shown in the following figure."

2

From this figure, the qubits around the plaquette and vertex operator can also take the eigenvalue -1 state. When actually constructing a Surface Code, is it the case that the stabilizer eigenvalue is not strictly 1, but that the eigenvalue -1 is also allowed?

Answer 1

Yes. The measured stabilizer value doesn't really matter as long as you perform the measurement and record the measurement value. This is because the codespace defined by all +1 eigenvalue of stabilizers has the same structure as another space defined by, e.g., all -1 eigenvalue.

Besides, the conclusion will not only apply to surface code, but to every stabilizer code when you wanna prepare the logical state with this kind of indrect syndrome measurement.