Quantum Gates

1. The Fundamental Unit: The Qubit

In classical systems, information is stored as bits (0 or 1). Quantum systems use the qubit (quantum bit).

Mathematical Representation

A qubit exists in a state vector or wave function denoted as $|\psi \rangle$. It is expressed as:

$\mid\psi\rangle = \alpha\mid0\rangle + \beta\mid1\rangle$

• Basis States: $\mid0\rangle$ and $\mid1\rangle$ are the fundamental discrete states
• Amplitudes: $\alpha$ and $\beta$ are complex numbers representing probability amplitudes.

Measurement and Duality

A qubit possesses a dual nature: it acts as an analog variable(any real number) before measurement (as $\alpha$ and $\beta$ vary continuously) but becomes digital(0/1) after measurement6.

• Superposition: Before being observed, the qubit exists in both states simultaneously This is illustrated by Schrödinger’s Cat, which is considered both “dead” and “alive” until the box is opened.
• Probability: The probability of collapsing to $\mid0\rangle$ is $|\alpha|^2$, and for $\mid1\rangle$ is $|\beta|^2$.
• Normalization: The total probability must always sum to 1 ($|\alpha|^2 + |\beta|^2 = 1$)10.

2. Visualizing States: The Bloch Sphere

The Bloch Sphere is a 3D geometric tool where any point on the surface represents a pure quantum state11.

• Z-Axis (Vertical): The North Pole is $\mid0\rangle = \begin{bmatrix} 1 \\ 0 \end{bmatrix}$ and the South Pole is $\mid1\rangle = \begin{bmatrix} 0 \\ 1 \end{bmatrix}$.

• X-Axis (Horizontal Equator): Represents the Hadamard Basis. $\mid+\rangle = \frac{\mid0\rangle + \mid1\rangle}{\sqrt{2}}$ is at the positive x-axis, and $\mid-\rangle = \frac{\mid0\rangle - \mid1\rangle}{\sqrt{2}}$ is at the negative x-axis.

• Y-Axis (Depth Equator): Includes complex phases. $\mid+i\rangle = \frac{\mid0\rangle + i\mid1\rangle}{\sqrt{2}}$ (positive y) and $\mid-i\rangle = \frac{\mid0\rangle - i\mid1\rangle}{\sqrt{2}}$ (negative y).

• General State: Defined by angles $\theta$ (latitude) and $\phi$ (longitude)15:

  $\mid\psi\rangle = \cos\frac{\theta}{2}\mid0\rangle + e^{i\phi}\sin\frac{\theta}{2}\mid1\rangle$

3. Quantum Gates

Quantum operations are represented by unitary matrices that rotate the state vector on the Bloch sphere.

Single-Qubit Gates

• X-Gate (Pauli-X): Acts as a bit-flip (the quantum NOT gate), mapping $\mid0\rangle \to \mid1\rangle$. It rotates the sphere 180° around the X-axis. e.g. If the input state is $\mid A \rangle = \alpha\mid0\rangle + \beta\mid1\rangle$, the output becomes $\mid B \rangle = \beta\mid0\rangle + \alpha\mid1\rangle$

$X= \begin{bmatrix} 0 &1 \\ 1 & 0 \end{bmatrix}$

• Z-Gate (Pauli-Z): A phase-flip gate. It leaves $\mid0\rangle$ unchanged but flips the sign of $\mid1\rangle$ ($\mid1\rangle \to -\mid1\rangle$). It rotates the sphere 180° around the Z-axis.

$Z= \begin{bmatrix} 1 &0 \\ 0 & -1 \end{bmatrix}$

• Hadamard (H) Gate: The most critical gate for superposition. it transforms basis states ($\mid0\rangle, \mid1\rangle$) into equatorial states ($\mid+\rangle, \mid-\rangle$).

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

• S and T Gates: Phase-shifting gates. The S-gate rotates 90° ($\pi/2$) maps $\mid +\rangle$ to $\mid i \rangle$and the T-gate rotates 45° ($\pi/4$) around the Z-axis.

$S= \begin{bmatrix} 1 &0 \\ 0 & i \end{bmatrix}$ $T= \begin{bmatrix} 1 &0 \\ 0 & \exp\left( \frac{i\pi}{4} \right) \end{bmatrix}$

Two Qubit Gates

• CNOT (Controlled NOT): Flips the second qubit only if the first qubit is $\mid1\rangle$. This is the primary tool for creating entanglement.

$CNOT = \begin{bmatrix} 1& 0 & 0 & 0\\0&1&0&0\\0&0&0&1\\0&0&1&0\end{bmatrix}$

• SWAP Gate: Exchanges the states of two qubits, often implemented using three CNOT gates. It used to move qubits under restricted connectivity.

$SWAP = \begin{bmatrix} 1& 0 & 0 & 0\\0&0&1&0\\0&1&0&0\\0&0&0&1\end{bmatrix}$

• Toffoli (CCNOT): A three-qubit gate where the third qubit flips only if the first two are both $\mid1\rangle$.

• Fredkin (CSWAP): Swaps the second and the thrid bit iff the control bit is $\mid1\rangle$

4. Governing Principles

Quantum computing is restricted by laws that do not apply to classical systems.

Reversibility

All quantum operations must be reversible; information cannot be destroyed. If you have the output state, you must be able to mathematically reconstruct the exact input state.

No-Cloning Theorem

It is physically impossible to create an identical, independent copy of an unknown quantum state.

• Entanglement vs. Copying: Using a CNOT gate on a qubit and a blank $\mid0\rangle$ qubit results in entanglement, not a clone.
• Teleportation: You can transfer a state to another qubit, but the original state is destroyed in the process