Pauli-X Gate (Bit-Flip) | Matrix, Dirac Notation & Rotation
Quick Technical Answer:
The Pauli-X gate is the quantum equivalent of the classical NOT gate. It flips a qubit from |0⟩ to |1⟩ and from |1⟩ to |0⟩, executing a π-radian (180°) rotation around the X-axis of the Bloch sphere.
Formula / Unitary:
X = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \quad X|0\rangle = |1\rangle, \quad X|1\rangle = |0\rangle
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Pauli-X Operator Dynamics
The Pauli-X matrix is involutory (X² = I) and unitary (X† = X = X⁻¹). It possesses eigenvalues λ = ±1 with corresponding eigenvectors |+⟩ = (|0⟩ + |1⟩)/√2 and |-⟩ = (|0⟩ - |1⟩)/√2.
Physical Implementation via Microwave RF Pulses
In superconducting transmon architectures, the Pauli-X gate is implemented by driving the qubit at its resonance frequency ω_01 with a calibrated π-pulse of duration t_gate ~ 20-40 ns using Derivative Removal by Adiabatic Gate (DRAG) pulsing to eliminate leakage into the |2⟩ state.
Pauli-X Truth Table
| Input State | Output State | Bloch Coordinates (Initial) | Bloch Coordinates (Final) |
|---|---|---|---|
| |0⟩ | |1⟩ | (0, 0, 1) | (0, 0, -1) |
| |1⟩ | |0⟩ | (0, 0, -1) | (0, 0, 1) |
| |+⟩ | |+⟩ | (1, 0, 0) | (1, 0, 0) |
| |-⟩ | -|-⟩ | (-1, 0, 0) | (-1, 0, 0) |
Frequently Asked Questions
Does the Pauli-X gate change the phase of a qubit?
The Pauli-X gate swaps the computational basis amplitudes without altering their relative signs on |0⟩ and |1⟩, but it introduces a global phase of -1 on the |-⟩ eigenstate.
How is a Pauli-X gate represented in OpenQASM?
In OpenQASM 2.0 and 3.0, the gate is invoked simply as `x q[0];`.