Quantum Gate

A basic operation on qubits, the quantum equivalent of classical logic gates.

A quantum gate is a basic operation that transforms quantum states. Just as classical computers use AND, OR, and NOT gates, quantum computers use quantum gates to perform computations.

Key Properties

Unitary

Every quantum gate is represented by a unitary matrix $U$: $$U^{†}U=U{U}^{†}=I$$

This ensures:

• Reversibility (every gate has an inverse: $U^{-1}=U^{†}$)
• Preservation of probability (states stay normalized)

Reversible

Unlike classical gates (AND, OR destroy information), quantum gates are always reversible. Given the output, you can recover the input.

Common Single-Qubit Gates

Common Multi-Qubit Gates

Gate Application

Applying gate $U$ to state $|\psi ⟩$: $$|{\psi }^{\prime }⟩=U|\psi ⟩$$

For density matrices: $${\rho }^{\prime }=U\rho U^{†}$$

Rotation Gates

Single-qubit gates can be parameterized as rotations:

$$R_x(\theta )=e^{-i\theta X/2},R_y(\theta )=e^{-i\theta Y/2},R_z(\theta )=e^{-i\theta Z/2}$$

Any single-qubit gate can be decomposed into rotations: $$U=e^{i\alpha }{R}_z(\beta )R_y(\gamma )R_z(\delta )$$

Gate Sets

A universal gate set can approximate any quantum operation:

• {H, T, CNOT}: standard universal set
• {H, Toffoli}: also universal

Physical Implementation

Gates are implemented differently on different hardware:

• Superconducting qubits: Microwave pulses
• Trapped ions: Laser pulses
• Photonic: Beam splitters, phase shifters

Gate times and error rates are key performance metrics. Fidelity measures how close a real gate is to the ideal operation.

See also: Quantum Circuit, Pauli Gates, Hadamard Gate, CNOT Gate, Universal Gate Set