# Quantum gate

**Quantum gates** are unitary operations that transform quantum states. Like classical logic gates (AND, OR, NOT), quantum gates manipulate qubits -- but they preserve superposition and enable entanglement. Every quantum gate is reversible (unitary matrix $U$ satisfies $U^\dagger U = I$).

A quantum gate on $n$ qubits is represented by a $2^n \times 2^n$ unitary matrix. It's an element of Lie group $\mathrm{SU}(2^n)$. Applying a gate to a state $|\psi\rangle$ gives a new state $U|\psi\rangle$. Gates compose: two gates in sequence is a new gate (matrix product).

Single-qubit gates act on one qubit; two-qubit gates entangle or correlate pairs; multi-qubit gates generalize. Any quantum computation can be decomposed into universal gate sets (e.g., single-qubit rotations + CNOT).

## Concepts

 1. [[quantum-gate-matrices|Gate matrices and notation]]
 2. [[quantum-gate-composition|Gate composition]]
 3. [[quantum-gate-decomposition|Gate decomposition]]
 4. [[quantum-gate-universal|Universal gate sets]]
 5. [[quantum-gate-parametric|Parametric gates]]
 6. [[quantum-gate-clifford|Clifford vs non-Clifford gates]]
 7. [[quantum-gate-physical-implementation|Physical implementation]]
 8. [[quantum-gate-fidelity|Gate fidelity and errors]]
 9. [[quantum-gate-stabilizer-formalism|Stabilizer formalism and simulation]]

## Gates

- [[quantum-gate-clifford]]
- [[quantum-gate-non-clifford]]
- [[quantum-gate-single-qubit]]
  - [[quantum-gate-pauli]]
    - [[quantum-gate-i|Identity (I)]]
    - [[quantum-gate-x|Pauli X (NOT)]]
    - [[quantum-gate-y|Pauli Y]]
    - [[quantum-gate-z|Pauli Z (π phase)]]
  - [[quantum-gate-h|Hadamard (H)]]
  - [[quantum-gate-phase]]
    - [[quantum-gate-p|P (θ phase)]]
    - [[quantum-gate-s|S (π/2 phase)]]
    - [[quantum-gate-t|T (π/4 phase)]]
    - [[quantum-gate-s-dagger|S† (inverse S gate)]]
    - [[quantum-gate-t-dagger|T† gate (inverse T gate)]]
  - [[quantum-gate-rotation]]
    - [[quantum-gate-rx|Rx (rotation around X)]]
    - [[quantum-gate-ry|Ry (rotation around Y)]]
    - [[quantum-gate-rz|Rz (rotation around Z)]]
    - [[quantum-gate-u|U (universal single-qubit)]]
- [[quantum-gate-two-qubit]]
  - [[quantum-gate-cx|CX (Controlled-NOT)]]
  - [[quantum-gate-cy|CY (Controlled-Y)]]
  - [[quantum-gate-cz|CZ (Controlled-Z)]]
  - [[quantum-gate-swap|SWAP]]
  - [[quantum-gate-iswap|iSWAP]]
  - [[quantum-gate-cp|CP (Controlled-Phase)]]
  - [[quantum-gate-xx|XX (Ising coupling)]]
  - [[quantum-gate-yy|YY (Ising coupling)]]
  - [[quantum-gate-zz|ZZ (Ising coupling)]]
- [[quantum-gate-three-qubit]]
  - [[quantum-gate-ccx|CCX (Toffoli)]]
  - [[quantum-gate-ccy|CCY (Controlled-Controlled-Y)]]
  - [[quantum-gate-ccz|CCZ (Controlled-Controlled-Z)]]
  - [[quantum-gate-cswap|CSWAP (Fredkin)]]
  - [[quantum-gate-ciswap|CiSWAP (Controlled-iSWAP)]]
  - [[quantum-gate-ccp|CCP (Controlled-Controlled-Phase)]]
  - [[quantum-gate-ccrx|CCRx (Controlled-Controlled-RX)]]
  - [[quantum-gate-ccry|CCRy (Controlled-Controlled-RY)]]
  - [[quantum-gate-ccrz|CCRz (Controlled-Controlled-RZ)]]
- [[quantum-gate-multiqubit|Multi-qubit gates (n-qubit general)]]
  - [[quantum-gate-permutation|Permutation gates]]
  - [[quantum-gate-grover-diffusion|Grover diffusion operator]]
  - [[quantum-gate-qft|Quantum Fourier Transform]]
  - [[quantum-gate-phase-oracle|Phase oracle gates]]