# Pauli X Gate (NOT)

**Pauli X** (also **NOT**) is the simplest quantum gate: it flips a qubit from $|0\rangle$ to $|1\rangle$ and vice versa.

Matrix:

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

Action: $X|0\rangle = |1\rangle$, $X|1\rangle = |0\rangle$.

## Properties

- **Self-inverse**: $X^2 = I$ (applying twice gives identity)
- **Eigenvalues**: $+1, -1$ with eigenvectors $(|0\rangle + |1\rangle)/\sqrt{2}$ and $(|0\rangle - |1\rangle)/\sqrt{2}$
- **Commutes with**: Y and Z anticommute with X
- **Classical analog**: classical bit NOT

## Uses

- **State preparation**: convert $|1\rangle$ to $|0\rangle$ or prepare superpositions
- **Error correction**: X errors are bit flips; syndrome measurements detect X errors
- **Algorithm building**: used in Grover's algorithm and many quantum circuits
- **Bloch sphere**: rotation by $\pi$ around x-axis

## Implementation

- **Superconducting qubits**: RX($\pi$) via full $\pi$ rotation around x-axis; gate time ~20–50 ns
- **Trapped ions**: resonant laser pulse at qubit transition frequency, tuned for $\pi$ rotation; gate time ~1–10 μs
- **Photonic**: optical rotation or interferometer configured for bit flip
- **Fidelity**: >99.9% on all platforms