# Is working with the |+> , |-> basis any harder than the |0>, |1> basis?

Say I have a code, for example the $$[[5,1,3]]$$ code, and I want to (fault tolerantly) prepare the logical $$|+ \rangle$$ state. Is that any harder than preparing the logical $$| 0 \rangle$$ state? Or what if I want to (fault tolerantly) measure in the logical $$|+ \rangle$$ basis (in other words measure in the basis consisting of the $$32$$ simultaneous eigenvectors of $$XXXXX, XZZXI, IXZZX, XIXZZ, ZXIXZ$$). Is that any harder than measuring in the logical $$| 0 \rangle$$ basis (in other words measuring in the basis consisting of the $$32$$ simultaneous eigenvectors of $$ZZZZZ, XZZXI, IXZZX, XIXZZ, ZXIXZ$$)?

In general my question is that when it comes to actually (fault tolerantly) implementing a quantum circuit is it any harder to work with (i.e. do state preparation and measurement) the $$|+ \rangle$$ basis as opposed to the the $$|0 \rangle$$ basis? For a code with transversal Hadamard it should be easy to fault tolerantly switch between the $$|+ \rangle$$ basis and the $$|0 \rangle$$ basis and so they should be equally easy to work with. But what about a code like the $$[[5,1,3]]$$ that doesn't have transversal Hadamard?

It entirely depends on the code you are using. For CSS codes, Z-basis and X-basis states are typically equally easy. They both have transversal initialization, the CNOT treats them symmetrically, etc.

• How about the $[[5,1,3]]$ code I gave in the example? Does that still have transversal initialization of some states? Could you maybe explain more about why transversal initialization of Z-basis and X-basis states always works for a CSS code? Commented Jul 4 at 22:16

$$[[5, 1, 3]]$$ is tri-symmetric for $$X,Y,Z$$ basis so I believe it won't showcase any bias for different basis choices.

For CSS codes (without transversal $$H$$), the answer seems quite apparent when your $$d_X\neq d_Z$$. A quick example is the repetition code where we only have Z stabilizers. Logical $$|0\rangle$$ is easier to prepare as it's just the tensor product of physical $$|0\rangle$$.

• What about a CSS code like the $[[15,1,3]]$ triorthogonal code? That does not have X,Y,Z symmetry from the Facet gate or X,Z symmetry from the Hadamard gate. Then how does working with $| + \rangle$ basis compare to working with $| 0 \rangle$ basis? Commented Jul 5 at 14:33