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Why do we need to reverse the order of qubits in Quantum Fourier Transform?

To understand why we need to reverse the qubit, let's take the formula given at the end of the part 4 of the tutorial you are pointing at. This formula is a rewrite of the QFT basis formula which is ...

Are the eigenvalues of projectors always zero and/or one?

The eigenvalues of projectors are always 0 or 1 (a projector satisfies $P^2=P$, which is equivalent to the eigenvalues satisfying $\lambda^2=\lambda$). This does not have any implication for equation (...
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I think that you can get $M'|\psi_1\rangle = |\psi_2\rangle$ by the following: $U(|0\rangle|\psi\rangle)$ is an eigenstate of $M$ with eigenvalue $1$ (just act on it with $M=U Z_1 U^\dagger$ to see ...