(6.2) Show that the operation $(2 \ket{\psi}\bra{\psi} - I)$ (where $\ket{\psi}$ is the equally weighted superposition of states) applied to general state $\sum_k \alpha_k \ket{k}$ produces

The quantum fourier transform on an orthonormal basis $\ket{0}, \cdots ,\ket{N - 1}$ is defined to be a linear operator with the following action on the basis states,

(4.1) Find the points on the Bloch sphere which correspond to the normalized eigenvectors of the different Paul matrices.

(11.8)

(2.11) Find the eigenvectors and eigenvalues of the Pauli matrices.