Qubits, Paulis, and Cliffords

This post is about qubits, the basic operations we perform on them, and how those operations generate powerful algebraic structures. We’ll meet the Pauli matrices, the Pauli group, and the Clifford group—central players in everything from quantum algorithms to error correction.

Qubit states and the Pauli matrices

In classical computing, the unit of information is the bit (0 or 1). In quantum computing, it’s the qubit. A single qubit lives in the two-dimensional complex vector space spanned by the basis states $|0\rangle$ and $|1\rangle$, and a general (pure) state is

\[|\psi\rangle=\alpha|0\rangle+\beta|1\rangle,\quad |\alpha|^2+|\beta|^2=1.\]

(Physical states are defined up to a global phase—multiplying by $e^{i\theta}$ doesn’t change measurement statistics.)

We represent the basis states as vectors:

\[|0\rangle=\begin{pmatrix}1\\0\end{pmatrix},\qquad |1\rangle=\begin{pmatrix}0\\1\end{pmatrix}.\]

Quantum operations (“gates”) are unitary matrices acting on these vectors. The most fundamental $2\times2$ matrices are the Paulis:

• Identity $I$: the “do nothing” gate.

  \[I=\begin{pmatrix}1&0\\0&1\end{pmatrix}\]

• Pauli-X $X$: the quantum NOT (bit-flip).

  \[X=\begin{pmatrix}0&1\\1&0\end{pmatrix}\]

• Pauli-Z $Z$: flips the phase of $|1\rangle$ (phase-flip).

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

• Pauli-Y $Y$: bit-flip plus phase, related by $Y=iXZ=-iZX$.

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

Key facts (for $X,Y,Z$):

• They’re Hermitian and unitary, hence $A^\dagger=A$ and $A^2=I$.
• Eigenvalues are $\pm1$.

• They anti-commute in pairs (e.g., $XZ=-ZX$) and multiply as

  \[XY=iZ,\quad YZ=iX,\quad ZX=iY \;\Rightarrow\; XYZ=iI.\]
• Together with $I$, they form an orthonormal basis (under the Hilbert-Schmidt inner product) for all $2\times2$ complex matrices.

The Pauli group $(P_1)$

The single-qubit Pauli group $P_1$ is the set of Paulis closed under multiplication, including global phases ${\pm1,\pm i}$:

\[P_1=\{\pm I,\pm iI,\pm X,\pm iX,\pm Y,\pm iY,\pm Z,\pm iZ\}.\]

The phases don’t change physics but make the set closed as a group.

Why it matters: Pauli operators form a basis for single-qubit errors. Any noise channel can be decomposed into (or approximated by) a mixture of Pauli errors—e.g., via Pauli twirling—which is a cornerstone of quantum error-correction analysis.

Key single-qubit gates

Beyond the Paulis, a few gates appear constantly.

Hadamard ($H$)

Creates and rotates superpositions:

\[H=\frac{1}{\sqrt2}\begin{pmatrix}1&1\\1&-1\end{pmatrix},\qquad H|0\rangle=|+\rangle,\; H|1\rangle=|-\rangle,\]

where $|{\pm}\rangle=(|0\rangle\pm|1\rangle)/\sqrt2 $. It’s its own inverse: $H^2=I$.

Phase ($S$)

A $\pi/2$ rotation about $Z$ (often “$\sqrt Z$”):

\[S=\begin{pmatrix}1&0\\0&i\end{pmatrix},\qquad S^2=Z.\]

(For universality you’ll also see the non-Clifford $T$ or “$\pi/8$” gate $T=\mathrm{diag}(1,e^{i\pi/4})$. It satisfies $T^2=S$ and, together with Clifford gates, yields universal quantum computation.)

The Clifford group: symmetries of the Paulis

The (single-qubit) Clifford group $C_1$ consists of all unitaries $V$ that normalize the Pauli group:

\[V P V^\dagger\in P_1\quad\text{for every }P\in P_1.\]

Intuitively, Clifford gates permute Pauli operators up to a sign.

Conjugation examples:

• $H X H^\dagger=Z, H Z H^\dagger=X$ (it swaps $X$ and $Z$).
• $S X S^\dagger=Y, S Z S^\dagger=Z$ (rotation around $Z$).

On $n$ qubits, the Clifford group is generated by $H$, $S$, and the two-qubit $\mathrm{CNOT}$ (below). Circuits built only from Clifford gates, Pauli measurements, and stabilizer-state preparation are efficiently simulable on a classical computer (the Gottesman–Knill theorem). To go beyond that power, you need at least one non-Clifford gate (e.g., $T$).

Two qubits and entanglement: the $\mathrm{CNOT}$ gate

To create entanglement we need multi-qubit gates. The workhorse is the $\mathrm{controlled-NOT}$ (control = first qubit, target = second). In the computational basis $|00\rangle,|01\rangle,|10\rangle,|11\rangle$,

\[\mathrm{CNOT}= \begin{pmatrix} 1&0&0&0\\ 0&1&0&0\\ 0&0&0&1\\ 0&0&1&0 \end{pmatrix}.\]

It flips the target iff the control is $|1\rangle$. $\mathrm{CNOT}$ is a Clifford gate and has clean error-propagation rules (conjugation on Pauli tensors):

\[\begin{aligned} \mathrm{CNOT}\,(X\!\otimes\! I)\,\mathrm{CNOT}^\dagger &= X\!\otimes\! X,\\ \mathrm{CNOT}\,(I\!\otimes\! X)\,\mathrm{CNOT}^\dagger &= I\!\otimes\! X,\\ \mathrm{CNOT}\,(Z\!\otimes\! I)\,\mathrm{CNOT}^\dagger &= Z\!\otimes\! I,\\ \mathrm{CNOT}\,(I\!\otimes\! Z)\,\mathrm{CNOT}^\dagger &= Z\!\otimes\! Z. \end{aligned}\]

In words: an $X$ (bit-flip) on the control propagates to the target; a $Z$ (phase-flip) on the target propagates back to the control. These identities are foundational in the design and analysis of stabilizer codes.

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What’s next: In the follow-up, we’ll look at the full $n$-qubit Clifford group, how big it is, and how Clifford+$T$ gives a universal fault-tolerant gate set.