Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 4 / 10C (Quantum Information and Computation)

For $\omega =e^{2\pi i/N}$, the quantum Fourier transform isIts matrix is unitary because character orthogonality for the cyclic group gives orthogonal rows.

For $N=4$, the binary label $10$ is the integer $2$, sowhere $|+⟩=(|0⟩+|1⟩)/\sqrt{2}$ and $|-⟩=(|0⟩-|1⟩)/\sqrt{2}$. It is a product state and hence has no quantum entanglement.

Applying the transform twice givesBy the finite geometric series, the inner sum over $y$ is $N$ when $x+z\equiv 0(\mathrm{mod}N)$ and zero otherwise. Thereforewhich is the stated $|N-x⟩$ with the label interpreted modulo $N$.

The map $x\mapsto -x$ is an involution, soIf $\lambda$ is an eigenvalue of ${\mathrm{QFT}}_N$, then ${\lambda }^4=1$. Hence every eigenvalue belongs toBecause $X^4-1$ has distinct roots, the transform is diagonalizable; which fourth roots occur and their multiplicities depend on $N$.