What are the logical 0 & 1 states for the 17 qubit (9 data qubit) surface code?

Question

What are the logical 1 and logical 0 states for the 17 qubit (9 data qubit ) surface code given stabilixzers below ?

I have read there is an algorithm that calculates the logical 0 and 1 states given the stabilizers . Any more info on this ?

Answer 1

The logical $|0\rangle_L$-state is created by initialising all data qubits in the $|0\rangle$-state, running all stabiliser circuits, and measuring all 8 stabilisers. You will end up in a random state, of which there are exponentially many, that respects the symmetries of the stabilisers. The $Z$-stabilisers measurement outcomes should all read 0, while the $X$-stabilisers all have a 50-50 probability of being measured as either 0 or 1.

The logical $|1\rangle_L$-state is created by applying the logical $X_L$ operator to the logical $|0\rangle_L$-state, in this case: $|1\rangle_L=X_L|0\rangle_L =X_2X_4X_6|0\rangle_L$. Note that initialising all data qubits in the $|1\rangle$-state and running the stabiliser circuits does NOT guarantee that you end up in the logical $|1\rangle_L$-state, for general stabiliser codes! In this case it does, but it's coincidental.

You should try this by hand, but you can show that the family of states corresponding to $|0\rangle_L$, when measured, will always give you a measurement outcome with an odd number of 0s, while measuring the $|1\rangle_L$-state will always give you an odd number of 1s. That is how you distinguish them in experiment.