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--- quantum-gate [June 12, 2026 at 22:00] – external edit 127.0.0.1
+++ quantum-gate [August 22, 2026 at 22:15] (current) – external edit 127.0.0.1
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-# Quantum gate
+# Quantum Gate
-**A quantum gate** is a unitary operation applied to one or more qubits. It is the quantum computing analog of a classical logic gate, but with two key differences: quantum gates are represented by matrices rather than truth tables, and they are always reversible — every quantum gate has an inverse.
-A quantum gate acting on $n$ qubits is a $2^n \times 2^n$ unitary matrix $U$, meaning $U^\dagger U = I$. Applying a gate to a state vector $\lvert\psi\rangle$ is a matrix-vector multiplication $U\lvert\psi\rangle$. Unitarity ensures that the total probability remains 1 after the operation. The reversibility of quantum gates follows directly from unitarity: the inverse of $U$ is $U^\dagger$, so every gate operation can be undone by applying the conjugate transpose.
+**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 $U^\dagger U = I$).
-Classical logic gates like AND and OR are irreversible — given only the output, you cannot recover both inputs. Quantum gates cannot be irreversible in this way because unitary evolution is bijective. The only irreversible step in a quantum computation is **measurement**, which collapses the state vector and cannot be undone.
+A quantum gate on $n$ qubits is represented by a $2^n \times 2^n$ unitary matrix. 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 are $2 \times 2$ unitary matrices. The most general single-qubit gate is the [[u-gate|U gate]] $U(\theta, \phi, \lambda)$, which subsumes all others as special cases. Two-qubit gates are $4 \times 4$ unitary matrices; the [[cnot-gate|CNOT gate]] is the standard entangling two-qubit gate. Together, single-qubit gates and CNOT form a universal gate set: any $n$-qubit unitary can be approximated to arbitrary precision using only these.
+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).
-## List of quantum gates
+## Concepts
- - [[pauli-gates]] — the four Pauli matrices $I, X, Y, Z$ as a group
+. [[quantum-gate-matrices|Gate matrices and notation]]
- - [[i-gate]] — identity
+. [[quantum-gate-composition|Gate composition]]
- - [[x-gate]] — bit flip (Pauli-X)
+. [[quantum-gate-decomposition|Gate decomposition]]
- - [[y-gate]] — bit and phase flip (Pauli-Y)
+. [[quantum-gate-universal|Universal gate sets]]
- - [[z-gate]] — phase flip (Pauli-Z)
+. [[quantum-gate-parametric|Parametric gates]]
- - [[h-gate]] — Hadamard
+. [[quantum-gate-clifford|Clifford vs non-Clifford gates]]
- - [[rotation-gates]] — rotation gates $R_x, R_y, R_z$
+. [[quantum-gate-physical-implementation|Physical implementation]]
- - [[p-gate]] — phase gate (generalizes $Z$, $S$, $T$)
+. [[quantum-gate-fidelity|Gate fidelity and errors]]
- - [[u-gate]] — universal single-qubit gate
- - [[multiqubit-gates]] — overview of two- and three-qubit gates
+## Gates
- - [[cnot-gate]] — controlled-NOT
- - [[toffoli-gate]] — doubly-controlled-NOT (CCNOT)
+- [[quantum-gate-single-qubit]]
- - [[swap-gate]] — qubit swap
+  - [[quantum-gate-i|Identity (I)]]
- - [[iswap-gate]] — swap with imaginary phase
+  - [[quantum-gate-x|Pauli X (NOT)]]
+  - [[quantum-gate-y|Pauli Y]]
+  - [[quantum-gate-z|Pauli Z]]
+  - [[quantum-gate-h|Hadamard (H)]]
+  - [[quantum-gate-s|S (Phase) gate]]
+  - [[quantum-gate-t|T gate]]
+  - [[quantum-gate-rx|RX (rotation around X)]]
+  - [[quantum-gate-ry|RY (rotation around Y)]]
+  - [[quantum-gate-rz|RZ (rotation around Z)]]
+  - [[quantum-gate-u|Universal single-qubit (U)]]
+- [[quantum-gate-two-qubit]]
+  - [[quantum-gate-cnot|CNOT (CX)]]
+  - [[quantum-gate-cz|CZ (Controlled-Z)]]
+  - [[quantum-gate-swap|SWAP]]
+  - [[quantum-gate-iswap|iSWAP]]
+  - [[quantum-gate-cphase|Controlled-Phase]]
+  - [[quantum-gate-xx|XX (parametric)]]
+  - [[quantum-gate-yy|YY (parametric)]]
+  - [[quantum-gate-zz|ZZ (parametric)]]
+- [[quantum-gate-three-qubit]]
+  - [[quantum-gate-toffoli|Toffoli (CCX)]]
+  - [[quantum-gate-fredkin|Fredkin (CSWAP)]]
+- [[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]]