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I am looking at a paper that consists of a stabilizer code that has only one stabilizer $$-c_1c_2c_3c_4$$. Here $$c_i$$ denote the hermitian majorana modes.

By the Jordan Wigner Transform the stabilizer becomes $$Z_1Z_2$$. So would I be correct in saying that it cannot protect against Z type errors?

I am not sure because I am still a bit new to Fermionic codes.

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It seems that we can indeed not use such a code to protect against Z type errors.

We know that the code protects against single qubit X errors, so protection against Z errors would imply that it can correct against arbitrary single qubit errors. But the code distance is 1, so this would be a contradiction.

The code distance is 1 because thereis 1 stabiliser and 2 qubits (1 majorana mode encodes 1 qubit as seen from the JW transform). So, the distance is 2-1 =1.

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