Flips |0⟩ ↔ |1⟩ for the target qubit.
The X gate is the quantum version of a classical NOT gate. It swaps the two basis states: if your qubit is in |0⟩, it becomes |1⟩, and vice versa. On the Bloch sphere, this is a 180° rotation around the X axis, flipping the north and south poles. But unlike classical NOT, which only works on definite bits, the X gate acts linearly on superpositions — it swaps the amplitudes of |0⟩ and |1⟩, preserving all quantum information.
Pauli-X is one of three Pauli operators (X, Y, Z) that form a basis for single-qubit operations. It is a unitary, Hermitian involution (X² = I) with eigenvalues ±1 and eigenstates |+⟩ and |−⟩. In the computational basis, it acts as a bit flip: X(α|0⟩ + β|1⟩) = α|1⟩ + β|0⟩. Combined with controlled operations, it becomes the fundamental building block for CNOT and Toffoli gates.
The measured bit result flips: 0 becomes 1 and 1 becomes 0.
Rotates 180° around the X axis. Swaps the north pole (|0⟩) and south pole (|1⟩). Equatorial states on the X axis (|+⟩, |−⟩) are unchanged.
The X gate looks simple, but it is not merely classical NOT. When acting on superpositions, it swaps amplitudes rather than flipping a definite bit, and this difference matters in interference.
X is the simplest non-trivial gate and the foundation of controlled operations. Understanding it fully — including how it interacts with superposition — is essential before moving to multi-qubit gates.
What is X|+⟩?