The Pauli gates: X, Y, Z

You’ve already clicked the X, Y, and Z buttons in the Hadamard lesson. Now we look at them properly. These three are the Pauli gates, named after Wolfgang Pauli. They are the three “natural” gates for a spin-$\frac{1}{2}$ system, and every other single-qubit gate you will ever see can be written as a combination of them.

If single-qubit gates are words, the Paulis are the letters.

The three gates

Here they are, with matrices and Bloch-sphere interpretations:

Pauli-X — the quantum bit flip:

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

$X$ swaps $|0\rangle$ and $|1\rangle$. Geometrically, it’s a rotation of $\pi$ around the $x$-axis of the Bloch sphere. It’s the quantum version of the classical NOT gate.

Pauli-Y — the “both at once” flip:

$Y = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}$

$Y$ also swaps $|0\rangle$ and $|1\rangle$, but it also multiplies the amplitudes by $\pm i$. Geometrically, it’s a rotation of $\pi$ around the $y$-axis. Notice that $Y = iXZ$ — it’s “X and Z combined,” with a global phase.

Pauli-Z — the phase flip:

$Z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$

$Z$ leaves $|0\rangle$ alone and multiplies $|1\rangle$ by $-1$. Looking at the computational-basis probabilities, nothing happens — $Z|0\rangle = |0\rangle$ and $Z|1\rangle = -|1\rangle$ (which is still $|1\rangle$ up to global phase). But on a superposition, $Z$ introduces a minus sign between the two components — and that relative phase is very visible. Geometrically, it’s a rotation of $\pi$ around the $z$-axis.

See them in action

Click through the three gates below. Start from $|0\rangle$. Try applying each Pauli on its own. Then try combinations.

Some things to notice as you play:

1. Apply $X$. The qubit flips from $|0\rangle$ to $|1\rangle$. Apply $X$ again — back to $|0\rangle$. So $X^2 = I$. Same for $Y^2 = I$ and $Z^2 = I$.
2. Start from $|0\rangle$ and apply $Z$. Nothing seems to change — the Bloch sphere still points at the north pole. That’s because $Z|0\rangle = |0\rangle$: the $|0\rangle$ state is an eigenstate of $Z$ with eigenvalue $+1$.
3. Apply $H$ first (making $|+\rangle$), then $Z$. Now look at the Bloch sphere — the arrow has moved from $|+\rangle$ to $|-\rangle$. The $Z$ gate, invisible on $|0\rangle$ alone, is very visible on superpositions.
4. Try $HXH$. You should get $Z$. Try $HZH$. You should get $X$. This is a crucial identity: the Hadamard swaps $X$ and $Z$.

A useful identity: anticommutation

The Pauli gates don’t commute with each other. They do something stronger — they anticommute:

$XY = -YX \qquad YZ = -ZY \qquad ZX = -XZ$

Each pair of Paulis, when you swap the order, picks up a minus sign. This is another way of saying that the three Bloch-sphere axes are mutually perpendicular — a rotation around $x$ followed by a rotation around $y$ is different from the other way around.

You can also write the three Pauli relations in a compact form: for any two different Paulis $P, Q \in \{X, Y, Z\}$,

$PQ = iR$

where $R$ is the third Pauli, following the cyclic order $X \to Y \to Z \to X$. For example, $XY = iZ$, $YZ = iX$, $ZX = iY$.

Why ‘Pauli’?

In 1925, Wolfgang Pauli wrote down three matrices to describe the spin of an electron — a genuinely quantum property with no classical analog. Those matrices are exactly $X$, $Y$, $Z$. The same three operators that describe a spin-$\frac{1}{2}$ particle’s behavior describe a qubit, because a qubit is — from a mathematical standpoint — a spin-$\frac{1}{2}$ system. The words “qubit” and “spin” are almost interchangeable if you squint.

What’s next

The Paulis are three “big steps” — each a rotation of exactly $\pi$. In Lesson 3.3 we meet the phase gates $S$ and $T$: smaller, more subtle rotations around the $z$-axis. Then in Lesson 3.4 we make those rotations continuous — you’ll get a slider and be able to rotate by any angle you like.