IQC 002

Ket

$|\psi\rangle = \begin{pmatrix} \psi_0 \\ \psi_1 \end{pmatrix}$

$|0\rangle = \begin{pmatrix} 1 \\ 0 \end{pmatrix}$

$|1\rangle = \begin{pmatrix} 0 \\ 1 \end{pmatrix}$

$|\psi\rangle = \psi_0 |0\rangle + \psi_1 |1\rangle \\ \quad = \psi_0 \begin{pmatrix} 1 \\ 0 \end{pmatrix} + \psi_1 \begin{pmatrix} 0 \\ 1 \end{pmatrix}$

Matrices

• Matrices represent linear transformations (quantum gates). A general 2x2 matrix is: $A = \begin{pmatrix} \alpha_{00} & \alpha_{01} \\ \alpha_{10} & \alpha_{11} \end{pmatrix}$
• Identity matrix: $I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$
• Linearity: $A(|\psi\rangle + |\phi\rangle) = A|\psi\rangle + A|\phi\rangle$

Bra and Daggers

Dagger

• Conjugate Transpose: swap rows/columns and complex-conjugates every entry.
• Ket becomes Bra: $|\psi\rangle^\dagger = \begin{pmatrix} \psi_0 \\ \psi_1 \end{pmatrix}^\dagger = \begin{pmatrix} \overline{\psi_0} & \overline{\psi_1} \end{pmatrix} = \langle\psi|$
• For a matrix: $A^\dagger = \begin{pmatrix} \overline{\alpha_{00}} & \overline{\alpha_{10}} \\ \overline{\alpha_{01}} & \overline{\alpha_{11}} \end{pmatrix}$
• Key Identities:
  • $(\alpha A)^\dagger = \overline{\alpha} A^\dagger$
  • $(A^\dagger)^\dagger = A$
  • $(AB)^\dagger = B^\dagger A^\dagger$

Hermitian and Unitary Matrices

• Hermitian: $H^\dagger = H$
  • Self-adjoint matrices, their eigenvalues are always real.
  • Observables in quantum mechanics are Hermitian.
• Unitary: $U^\dagger U = UU^\dagger = I$
  • The inverse of a unitary matrix is its conjugate transpose.
  • Unitary matrices preserve norms

Inner Product

• The angle between two vectors $|\psi\rangle$ and $|\phi\rangle$ is defined as bra-ket: $\langle\psi|\phi\rangle = (\overline{\psi_0} \overline{\psi_1}) \begin{pmatrix} \phi_0 \\ \phi_1 \end{pmatrix} = \overline{\psi_0}\phi_0 + \overline{\psi_1}\phi_1$
• Important properties:
  • Order Matters: $\langle\psi|\phi\rangle \neq \langle\phi|\psi\rangle$
  • But $\langle\psi|\phi\rangle = \overline{\langle\phi|\psi\rangle}$ (Complex Conjugate)
  • The modulus is symmetric: $|\langle\psi|\phi\rangle| = |\langle\phi|\psi\rangle|$
• The Magnitude of a vector is given by: $\| |\psi\rangle \|^2 = \langle\psi|\psi\rangle = |\psi_0|^2 + |\psi_1|^2$

Orthonormal of the Computational Basis

• The basis states $|0\rangle$ and $|1\rangle$ are orthonormal: $\langle 0|0\rangle = 1, \quad \langle 1|1\rangle = 1, \quad \langle 0|1\rangle = 0, \quad \langle 1|0\rangle = 0$
• This simplifies inner products enormously when working with the computational basis: $\langle\psi|\phi\rangle = (\overline{\psi_0}\langle0| + \overline{\psi_1}\langle1|) (\phi_0|0\rangle + \phi_1|1\rangle)$
• All corss terms vanish due to orthogonality, leaving: $= \overline{\psi_0}\phi_0 \langle0|0\rangle + \overline{\psi_0}\phi_1 \langle0|1\rangle + \overline{\psi_1}\phi_0 \langle1|0\rangle + \overline{\psi_1}\phi_1 \langle1|1\rangle$ $= \overline{\psi_0}\phi_0 + \overline{\psi_1}\phi_1$

Outer Products

• The outer product of two vectors produces a matrix: $|\psi\rangle\langle\phi| = \begin{pmatrix} \psi_0 \\ \psi_1 \end{pmatrix} \begin{pmatrix} \overline{\phi_0} & \overline{\phi_1} \end{pmatrix} = \begin{pmatrix} \psi_0\overline{\phi_0} & \psi_0\overline{\phi_1} \\ \psi_1\overline{\phi_0} & \psi_1\overline{\phi_1} \end{pmatrix}$
• Basis outer products: $|0\rangle\langle1| = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}, \quad |1\rangle\langle0| = \begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix}$
• Any matrix can be expanded in terms of outer products of the computational basis: $A = \alpha_{00} |0\rangle\langle0| + \alpha_{01} |0\rangle\langle1| + \alpha_{10} |1\rangle\langle0| + \alpha_{11} |1\rangle\langle1|$

The Qubit

A qubit is the fundamental unit of quantum information

$|\psi\rangle = \alpha |0\rangle + \beta |1\rangle$

• where $\alpha, \beta \in \mathbb{C}$ are complex numbers such that $|\alpha|^2 + |\beta|^2 = 1$ (normalization condition).
• Any normalized single-qubit state can be parameterized using two angles $\theta$ and $\phi$ (real numbers): $|\psi\rangle = \cos\theta|0\rangle + e^{i\phi}\sin\theta|1\rangle$
• Key difference from a bit: a bit is either 0 or 1, while a qubit can be in a superposition of both states simultaneously until measured.

Measurement

• When you measure a qubit $|\psi\rangle = \alpha |0\rangle + \beta |1\rangle$ in the computational basis, you get:
  • $|0\rangle$ with probability $|\alpha|^2$
  • $|1\rangle$ with probability $|\beta|^2$

Outcome	Probability	Post-measurement State
0	$	\alpha
1	$	\beta

• The result of measuring a qubit is a single classical bit.
• For $|\psi\rangle = \cos\theta|0\rangle + e^{i\phi}\sin\theta|1\rangle$:
  • Probability of measuring $|0\rangle$: $\cos^2\theta$
  • Probability of measuring $|1\rangle$: $\sin^2\theta$
  • The phase $\phi$ does not affect measurement outcomes.

One-Qubit Gates

Pauli Matrices

Unitary matrics

$\mathbb{I} = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}, \quad X = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \quad Y = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}, \quad Z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$

• $X$ is the quantum NOT gate:
  • $X|0\rangle = |1\rangle$
  • $X|1\rangle = |0\rangle$
• $Z$ filps the phase of $|1\rangle$:
  • $Z|0\rangle = |0\rangle$
  • $Z|1\rangle = -|1\rangle$
• All three $(X, Y, Z)$ are both Hermitian and unitary.

Hadamard Gate

$H = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix}$

• $H$ creates superpositions:
  • $H|0\rangle = \frac{|0\rangle + |1\rangle}{\sqrt{2}}$
  • $H|1\rangle = \frac{|0\rangle - |1\rangle}{\sqrt{2}}$
• $H$ also "un-does" superpositions:
  • $H\left(\frac{|0\rangle + |1\rangle}{\sqrt{2}}\right) = |0\rangle$
  • $H\left(\frac{|0\rangle - |1\rangle}{\sqrt{2}}\right) = |1\rangle$

Rotation Gate

$R(\theta) = \begin{pmatrix} \cos\theta & \sin\theta \\ -\sin\theta & \cos\theta \end{pmatrix}$

• $R(\theta)$ rotates the state vector by an angle $\theta$ in the $|0\rangle$-$|1\rangle$ plane.

The Bloch Sphere

Every single-qubit state $|\psi\rangle = \cos\theta|0\rangle + e^{i\phi}\sin\theta|1\rangle$ maps to a point on the surface of a unit sphere

• $\theta$ is polar angle from north pole
• $\phi$ is azimuthal angle around equator
• $|0\rangle$ is at the north pole
• $|1\rangle$ is at the south pole
• $\frac{|0\rangle + |1\rangle}{\sqrt{2}}$ is on the equator at $\phi=0$

Exercises

1. For any two-dimensional state vector $|\psi\rangle = \alpha |0\rangle + \beta |1\rangle$, it holds that $|\alpha|^2 + |\beta|^2 = 1$.
2. Measuring a qubit in the $\{|0\rangle, |1\rangle\}$ basis yields a probabilistic outcome when both $\alpha$ and $\beta$ are non-zero.
3. A global phase factor $e^{i\gamma}$ applied to a qubit state does not change the probabilities of measurement outcomes in the computational basis.
4. The Bloch sphere represents all pure single-qubit states as points on the surface of the sphere.
5. A real $2 \times 2$ matrix is a valid quantum gate only if it is unitary, not merely invertible.
6. The Pauli-X gate flips $|0\rangle$ to $|1\rangle$ and $|1\rangle$ to $|0\rangle$.
7. When a qubit is measured in a given basis, the state collapses to the basis state corresponding to the measurement outcome.
8. Unitary matrices preserve the norm of any vector they act on.
9. If $|\psi\rangle = \cos(\theta)|0\rangle + e^{i\phi}\sin(\theta)|1\rangle$, then the probability of measuring $|0\rangle$ is $\cos^2(\theta)$.
10. The state $\frac{1}{2}|0\rangle + \frac{\sqrt{3}}{2}|1\rangle$ is properly normalized.
11. If $|\psi\rangle = \frac{1}{\sqrt{3}}|0\rangle + \sqrt{\frac{2}{3}} e^{i\pi/4}|1\rangle$, then the probability of measuring $|0\rangle$ is $\frac{1}{3}$.
12. The states $\frac{1}{\sqrt{2}}(|0\rangle + |1\rangle)$ and $\frac{1}{\sqrt{2}}(|0\rangle - |1\rangle)$ are orthogonal.
13. The Pauli-X matrix $X = \begin{pmatrix}0 & 1 \\ 1 & 0\end{pmatrix}$ satisfies $X^2 = I$, where $I$ is the $2 \times 2$ identity matrix.
14. Applying the Pauli-Z gate $Z = \begin{pmatrix}1 & 0 \\ 0 & -1\end{pmatrix}$ to $\frac{1}{\sqrt{2}}(|0\rangle + i|1\rangle)$ produces $\frac{1}{\sqrt{2}}(|0\rangle - i|1\rangle)$.