Pauli-X Gate (Quantum NOT)
Flips |0⟩ to |1⟩ and vice versa — the quantum equivalent of a classical NOT gate.
Formula
Variables
Example
X|0⟩ = |1⟩, X|1⟩ = |0⟩
How it works
The Pauli-X gate is a 2×2 matrix that swaps the amplitudes of the basis states |0⟩ and |1⟩. If a qubit is written as a column vector [a, b]^T, multiplying by X gives [b, a]^T, so the probability amplitudes are exchanged and the bit value is effectively flipped. This works because matrix multiplication acts linearly on the state vector, and X is designed to implement the simplest possible reversible bit flip in quantum mechanics.
When to use it
Use the Pauli-X gate whenever you need a quantum circuit to invert a qubit state, prepare a specific basis state from a known starting state, or build larger logic operations from simple reversible pieces. It is one of the first gates used in quantum computing because it mirrors the classical NOT operation while still fitting the rules of quantum states.
Worked example, step by step
Start with the qubit state |ψ⟩ = 0.6|0⟩ + 0.8|1⟩ Write it as a vector: |ψ⟩ = [0.6, 0.8]^T Apply the Pauli-X matrix: X|ψ⟩ = [[0, 1], [1, 0]] [0.6, 0.8]^T Multiply the matrix and vector: X|ψ⟩ = [0·0.6 + 1·0.8, 1·0.6 + 0·0.8]^T X|ψ⟩ = [0.8, 0.6]^T So the new state is 0.8|0⟩ + 0.6|1⟩
Common mistakes to avoid
- Thinking X changes the overall probability of the state instead of swapping the amplitudes
- Forgetting that a qubit can be in a superposition, so X acts on both components at once
- Mixing up the matrix entries and writing the off-diagonal 1s in the wrong positions
Did You Know?
The three Pauli gates (X, Y, Z) correspond to 180° rotations around the three axes of the Bloch sphere.
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