Pauli Gates

The three fundamental single-qubit gates (X, Y, Z) that form the building blocks of quantum operations.

The Pauli gates (also called Pauli matrices or Pauli operators) are three fundamental single-qubit operations named after physicist Wolfgang Pauli. They’re ubiquitous in quantum computing.

The Three Gates

Pauli X (Bit Flip)

$$X=\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)$$

• Swaps $|0⟩\leftrightarrow |1⟩$
• Quantum analog of classical NOT
• 180° rotation around X-axis on Bloch sphere

$$X|0⟩=|1⟩,X|1⟩=|0⟩$$

Pauli Y

$$Y=\left(\begin{array}{cc}0& -i\\ i& 0\end{array}\right)$$

• Combined bit and phase flip
• 180° rotation around Y-axis

$$Y|0⟩=i|1⟩,Y|1⟩=-i|0⟩$$

Pauli Z (Phase Flip)

$$Z=\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right)$$

• Leaves $|0⟩$ unchanged, flips sign of $|1⟩$
• 180° rotation around Z-axis

$$Z|0⟩=|0⟩,Z|1⟩=-|1⟩$$

Key Properties

Self-Inverse

Each Pauli gate is its own inverse: $$X^2=Y^2=Z^2=I$$

Anti-Commutation

Paulis anti-commute with each other: $$XY=-YX=iZ$$ $$YZ=-ZY=iX$$ $$ZX=-XZ=iY$$

Hermitian

All Paulis are Hermitian ($P=P^{†}$), meaning they’re also valid observables (measurement operators).

Eigenvalues

Each Pauli has eigenvalues +1 and -1:

• X eigenstates: $|+⟩,|-⟩$
• Y eigenstates: $|i⟩,|-i⟩$
• Z eigenstates: $|0⟩,|1⟩$

Pauli Group

The Pauli group consists of all products of Paulis with phases $\{\pm 1,\pm i\}$: $${\mathcal{P}}_1=\{\pm I,\pm iI,\pm X,\pm iX,\pm Y,\pm iY,\pm Z,\pm iZ\}$$

For $n$ qubits, Pauli strings are tensor products like $X\otimes Z\otimes Y$.

Why They Matter

Paulis are fundamental because:

• They form a basis for all 2×2 matrices
• Quantum error correction describes errors as Pauli operations
• Pauli decomposition expresses Hamiltonians in terms of Paulis
• Measurements are often in Pauli bases (X, Y, or Z)

See also: Quantum Gate, Bloch Sphere, Hadamard Gate