(i) What is the action of $\mathrm{QFT}_{N}$ on a state $|x\rangle$, where $x \in\{0,1,2, \ldots, N-1\}$ and $\mathrm{QFT}_{N}$ denotes the Quantum Fourier Transform modulo $N$ ?

(ii) For the case $N=4$ write $0,1,2,3$ respectively in binary as $00,01,10,11$ thereby identifying the four-dimensional space as that of two qubits. Show that $\mathrm{QFT}_{N}|10\rangle$ is an unentangled state of the two qubits.

(iii) Prove that $\left(\mathrm{QFT}_{N}\right)^{2}|x\rangle=|N-x\rangle$, where $\left(\mathrm{QFT}_{N}\right)^{2} \equiv \mathrm{QFT}_{N} \circ \mathrm{QFT}_{N}$.

[Hint: For $\omega=e^{2 \pi i / N}, \sum_{m=0}^{N-1} \omega^{m K}=0$ if $K$ is not a multiple of $N$.]

(iv) What is the action of $\left(\mathrm{QFT}_{N}\right)^{4}$ on a state $|x\rangle$, for any $x \in\{0,1,2, \ldots, N-1\}$ ? Use the above to determine what the eigenvalues of $\mathrm{QFT}_{N}$ are.