Pauli-X Gate

Overview

The first named gate to learn is Pauli-X (often just “X”).

It is the closest single-qubit gate to a classical NOT:

• it swaps $|0\rangle$ and $|1\rangle$.

But because it is a quantum gate, it also acts sensibly on superpositions. This page builds both the basis-state intuition and the Bloch-sphere picture.

Intuition

In the Bloch-sphere picture:

• $|0\rangle$ is the +$z$ direction (north pole).
• $|1\rangle$ is the −$z$ direction (south pole).

The X gate flips those two poles. So it acts like a 180° rotation that sends +$z$ to −$z$.

It also flips any state’s Bloch vector accordingly. If a state points somewhere on the sphere, X rotates it to the opposite point across the $x$-axis rotation.

A key idea:

• gates don’t just change probabilities; they can change the direction of a superposition on the sphere.

Formal Description

Action on basis states

The defining action is

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

By linearity, this already tells you what X does to any superposition. For example, if

$$|\psi\rangle = \alpha|0\rangle + \beta|1\rangle,$$

then

$$X|\psi\rangle = X(\alpha|0\rangle + \beta|1\rangle) = \alpha X|0\rangle + \beta X|1\rangle = \alpha|1\rangle + \beta|0\rangle.$$

So X swaps the amplitudes associated with $|0\rangle$ and $|1\rangle$.

Minimal matrix form (introduced gently)

If we write states as coordinate vectors in the computational basis, X can be represented by a 2×2 matrix.

We choose the basis vectors as

$$|0\rangle \equiv \begin{pmatrix}1\\0\end{pmatrix},\qquad |1\rangle \equiv \begin{pmatrix}0\\1\end{pmatrix}.$$

The unique matrix that sends $|0\rangle$ to $|1\rangle$ and $|1\rangle$ to $|0\rangle$ is

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

Read the matrix as “swap the two components.”

Worked Example

Take the state

$$|\psi\rangle = \tfrac{3}{5}|0\rangle + \tfrac{4}{5}|1\rangle.$$

Apply X:

$$X|\psi\rangle = \tfrac{3}{5}|1\rangle + \tfrac{4}{5}|0\rangle = \tfrac{4}{5}|0\rangle + \tfrac{3}{5}|1\rangle.$$

So the probabilities in the computational basis are swapped:

• before: $P(0)=9/25$, $P(1)=16/25$
• after: $P(0)=16/25$, $P(1)=9/25$

Bloch-sphere interpretation:

• X flips the state across a 180° rotation about the $x$ axis.

Turtle Tip

Common Pitfalls

Quick Check

What’s Next

X is the “bit-flip-like” Pauli gate. Next we study Pauli-Z, which leaves computational-basis probabilities unchanged but changes relative phase—a deeply quantum effect.