How to verify that a certain gate was applied to a quantum code

Question

Suppose I have a quantum error correcting code $|\psi \rangle = \alpha | 0 \rangle + \beta | 1 \rangle$, say the $[[7,1,3]]$ Steane code for concreteness.

Suppose there is a black box that either implements logical phase $\overline{S}$ or logical phase conjugate $\overline{S}^\dagger$ so that we are either in the state $\overline{S} | \psi \rangle$ or the state $\overline{S}^\dagger |\psi \rangle$.

Is there any way to reliably (and not destructively) verify which gate was actually applied to the code?

Answer 1

Let's define $|\tilde\psi\rangle=S^\dagger|\psi\rangle$. Then you're asking the simply question of whether we can distinguish between $|\tilde\psi\rangle$ and $S^2|\tilde\psi\rangle=Z|\tilde\psi\rangle$. So if $|\tilde\psi\rangle=|+\rangle$, we'd have to distinguish between the + and - states, which one can do with an $X$ measurement. So, now we just have to work it backwards: fix $|\psi\rangle=S^\dagger|+\rangle$, pass it into the oracle, and measure in the $X$ basis.