Quantum Computing (IT-802 (D)) - Important Questions

Define a unitary operator. Prove that an operator $U$ on a finite-dimensional Hilbert space is unitary if and only if $U^{\dagger}U=UU^{\dagger}=I$. Show that if $U$ is unitary then every eigenvalue $\lambda$ of $U$ satisfies $|\lambda|=1$. Give an explicit example of a $2\times 2$ unitary matrix, for example the Hadamard matrix $$H=\frac{1}{\sqrt{2}}\left(\begin{array}{cc}1 & 1\\1 & -1\end{array}\right)$$ and compute its eigenvalues and normalized eigenvectors.

Define a projector (projection operator) $P$. Prove that $P$ is a projector iff $P^{2}=P$ and $P^{\dagger}=P$. Explain how projectors represent projective measurements in quantum mechanics. For the qubit state $|\psi\rangle=\alpha|0\rangle+\beta|1\rangle$ with $|\alpha|^{2}+|\beta|^{2}=1$, let $P=|0\rangle\langle 0|$. Calculate the probability of obtaining the outcome associated with $P$ and determine the post-measurement state.

State and prove the spectral theorem for normal operators on a finite-dimensional Hilbert space. Use the theorem to show that any Hermitian operator $H$ can be diagonalized by a unitary $U$ such that $$H=U^{\dagger}DU$$ where $D$ is a real diagonal matrix. Explain why eigenvalues of $H$ are real and eigenvectors corresponding to distinct eigenvalues are orthogonal.

Compute the eigenvalues and normalized eigenvectors of the Pauli matrices $\sigma_{x}$, $\sigma_{y}$ and $\sigma_{z}$. Explicitly diagonalize $\sigma_{x}$ by finding a unitary $U$ such that $$U^{\dagger}\sigma_{x}U\;=\;\text{(diagonal)}\,.$$

Show that any unitary operator $U$ preserves inner products and norms: for arbitrary vectors $|\phi\rangle$ and $|\psi\rangle$ prove that $$\langle\phi|\psi\rangle=\langle U\phi|U\psi\rangle$$ and $$\|U|\psi\rangle\|=\||\psi\rangle\|.$$ Give a short example illustrating these properties for a single-qubit unitary.