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I'm trying to define a cylindrical surface code; that is, a surface code that has boundaries in 1 axis and that is periodic over the other. Let's assume it is periodic in the horizontal axis, and it has rough boundaries on top and bottom.

Now, if I understood well this homology equivalence thing. We could define a logical operator X from top to bottom. And the periodicity on the horizontal axis, should allow us to define one X and one Z logical operator right (since we can wrap around the cylinder with both)?

If this is correct (which I'm not sure), we have a setup with 2 logical X and only 1 logical Z. What does that correspond to in a qubits perspective (like would it possibly define some kind of qutrit)?

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The second "logical X" is a product of stabilizers, so not actually a logical operator.

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