Quantum Gate

quantum-computing

Definition

Quantum Gate

A quantum gate is a unitary operator that acts on one or more qubits to transform the quantum state of a quantum register. Quantum gates are the building blocks of quantum algorithms, representing reversible operations that preserve the norm of the state vector.

1. Unitarity: Every quantum gate $U$ satisfies $U^{†}U=U{U}^{†}=I$.
2. Reversibility: All quantum gates are invertible; the inverse is $U^{†}$.
3. Linearity: Gates act linearly on superpositions: $U(\alpha |\psi ⟩+\beta |\varphi ⟩)=\alpha U|\psi ⟩+\beta U|\varphi ⟩$.

Common Single-Qubit Gates

Pauli-X Gate (NOT):

$$X=\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)$$

Hadamard Gate:

$$H=\frac{1}{\sqrt{2}}\left(\begin{array}{cc}1& 1\\ 1& -1\end{array}\right)$$

Pauli-Z Gate:

$$Z=\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right)$$

Two-Qubit Gates

CNOT (Controlled-NOT):

$$\text{CNOT}=\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 0& 1\\ 0& 0& 1& 0\end{array}\right)$$

Applies $X$ to the target qubit if the control qubit is $|1⟩$.