Problem 3: Proving the correctness of the phase flip code
- What are the eigenvalue/eigenvector pairs of the \(X\) operator? If we perform a projective measurement using this operator, which basis of measurement are we representing?
- What are the eigenvalue/eigenvector pairs of the \(XX\) operator? Let \(P_+\) be the state representing the projection onto the +1 eigenspace and \(P_-\) be the state representing the projection onto the -1 eigenspace of the \(XX\) operator.
- Consider a state \(a\left|0\right> + b\left|1\right>\) encoded using the phase flip code: \(a\left|+++\right> + b\left|---\right>\). Write down the state after a phase flip error occurs on the last qubit.
- Take the state from the previous problem, and calculate the measurement of the \(X_1X_2\) observable followed by a measurement of the \(X_2X_3\) observable. What is the sequence of eigenvalues observed?
- What is the state after these measurements? Did we learn anything about the amplitudes \(a\) and \(b\)?