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

Define a qubit. Represent a general single-qubit state in the computational basis $|\psi\rangle = \alpha|0\rangle + \beta|1\rangle$ and state the normalization condition for $\alpha$ and $\beta$.

Show that any pure single-qubit state can be written as $|\psi\rangle = \cos\left(\frac{\theta}{2}\right)|0\rangle + e^{i\phi}\sin\left(\frac{\theta}{2}\right)|1\rangle$. Explain the geometric meaning of the angles $\theta$ and $\phi$ on the Bloch sphere.

Explain the concept of quantum superposition. Give an example of a single-qubit superposed state and contrast it with a classical probabilistic mixture of $|0\rangle$ and $|1\rangle$.

State the postulates of quantum measurement for a single qubit. For the state $|\psi\rangle = \alpha|0\rangle + \beta|1\rangle$, compute the probabilities of obtaining outcomes $0$ and $1$ when measured in the computational basis.

Explain how multi-qubit states are constructed using the tensor product. Write the two-qubit product state $|\psi\rangle = |0\rangle\otimes\left(\alpha|0\rangle + \beta|1\rangle\right)$ explicitly and provide an example of a two-qubit entangled state.

Define quantum entanglement for a two-qubit system. Show that the Bell state $|\Phi^{+}\rangle = \frac{1}{\sqrt{2}}\left(|00\rangle + |11\rangle\right)$ is not separable by attempting to express it as a tensor product of single-qubit states.

Explain the density matrix formalism for a single qubit. Distinguish between pure and mixed states. Show that any single-qubit density matrix can be written as $$\rho = \frac{1}{2}\left(I + \vec{r}\cdot\vec{\sigma}\right)$$ and state the constraint on the Bloch vector $\vec{r}$ for physical states.

Write the matrix representations of the Pauli gates $X$, $Y$, $Z$ and the Hadamard gate $H$. For each gate, show its action on the computational basis states $|0\rangle$ and $|1\rangle$.