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

Define a qubit. Express a general single-qubit state in bra-ket notation and state the normalization condition. Write $\left|0\right\rangle$ and $\left|1\right\rangle$ in column-vector form.

Explain the Bloch sphere representation of a single qubit. Show that any pure single-qubit state can be written as

$\left|\psi\right\rangle = \cos\left(\frac{\theta}{2}\right)\left|0\right\rangle + e^{i\phi}\sin\left(\frac{\theta}{2}\right)\left|1\right\rangle$.

Give the correspondence between $\theta,\phi$ and the Bloch vector components $\left( x,y,z\right)$.

Describe multi-qubit states and the tensor-product construction. For two qubits, write the computational basis states and express a general two-qubit state $\left|\Psi\right\rangle$.

Differentiate between separable (product) states and entangled states with one explicit example of each.

Using bra-ket notation, define the inner product $\left\langle\phi\middle|\psi\right\rangle$ and the outer product $\left|\psi\right\rangle\left\langle\phi\right|$. Show the matrix forms of $\left|0\right\rangle$, $\left|1\right\rangle$, $\left\langle 0\right|$, and $\left\langle 1\right|$, and compute $\left|0\right\rangle\left\langle 0\right|$ and $\left|0\right\rangle\left\langle 1\right|$.

Explain how single-qubit unitary gates correspond to rotations on the Bloch sphere. Write the Pauli matrices $\sigma_x$, $\sigma_y$, $\sigma_z$ and the rotation operator about a unit vector $\hat{n}$:

$R_{\hat{n}}\left(\theta\right) = e^{-i\frac{\theta}{2}\,\hat{n}\cdot\vec{\sigma}}$.

Describe the action of $R_{\hat{y}}\left(\pi\right)$ on the computational basis states.

Describe measurement of a single qubit in the computational basis $\left\{\left|0\right\rangle,\left|1\right\rangle\right\}$. If $\left|\psi\right\rangle = \alpha\left|0\right\rangle + \beta\left|1\right\rangle$, state the probabilities of outcomes and the post-measurement states.

Define the Bell state $\left|\Phi^{+}\right\rangle = \dfrac{1}{\sqrt{2}}\left(\left|00\right\rangle + \left|11\right\rangle\right)$. Show that $\left|\Phi^{+}\right\rangle$ cannot be written as a tensor product of two single-qubit states (i.e., prove it is entangled).