Pauli gates (I, X, Y, Z) form the fundamental set of single-qubit gates that generate the Pauli group. They are Clifford gates and commute/anticommute in well-defined ways, making them essential for quantum error correction and circuit decomposition.

• Identity (I): no-op; eigenvalues ±1
• Pauli X (NOT): bit flip around x-axis
• Pauli Y: rotation around y-axis; composition of X and Z
• Pauli Z: phase flip; $Z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$

• Involution: All Paulis are self-inverse ($P^2 = I$ for all P)
• Anticommutation: $\{P_i, P_j\} = 0$ for $i \neq j$ (any two distinct Paulis anticommute)
• Eigenvalues: All Paulis have eigenvalues ±1
• Completeness: Any single-qubit unitary can be decomposed as a linear combination of Paulis

• Error correction: Syndrome measurements detect which Pauli error occurred
• Tomography: Measuring expectations of all three X, Y, Z gives complete state info
• Circuit synthesis: Arbitrary unitaries decompose into Pauli rotations (RX, RY, RZ)