For this code, we'll detect $X$ and $Z$ errors separately, similarly to how we did it for bit flip code and phase flip code, and then combine the results into the final answer. A $Y$ error will be signaled by both $X$ and $Z$ errors occurring, since the problem states that at most one error happened.

1. To detect $X$ errors, we recall that in Shor code, each triplet of qubits $0 \ldots 2, 3 \ldots 5, 6 \ldots 8$ is an encoding of a $\ket{+}$ state in the bit flip code. This means that we can detect an $X$ error by applying the bit flip error detection logic to each triplet of qubits separately. 

2. To detect $Z$ errors, we recall that 6-qubit joint measurements in $X$ basis can detect the parity of the relative phase of the first three qubits and the last three qubits in the measurement, that is, distinguish

    $$\frac12 \left( (\ket{000} + \ket{111}) \otimes (\ket{000} + \ket{111}) + (\ket{000} - \ket{111}) \otimes (\ket{000} - \ket{111}) \right)$$

    from 

    $$\frac12 \left( (\ket{000} + \ket{111}) \otimes (\ket{000} - \ket{111}) + (\ket{000} - \ket{111}) \otimes (\ket{000} + \ket{111}) \right)$$

    This means that we can detect a $Z$ error by doing two 6-qubit joint measurements on qubit triplets $0 \ldots 2$ & $3 \ldots 5$ and on $3 \ldots 5$ & $6 \ldots 8$ and interpret their results in the same way as we did for bit flip and phase flip codes.

3. To combine the results, we check which of the results occurred: 

- if neither $X$ nor $Z$ error was detected, we return "no error";
- if both were detected, we return the $Y$ error and use the index of the qubit where the $X$ error was detected (remember that we can tract the $Z$ error only to the triplet in which it occurred, not to the exact qubit);
- if only one error was detected, we return that error.

@[solution]({
    "id": "qec_shor__shor_detect_solution",
    "codePath": "Solution.qs"
})
