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

• 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

• 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

• 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