Pauli Operators
The four fundamental 2x2 matrices acting on a single qubit: I, X, Y, Z.
- I — identity, does nothing
- X — bit-flip (|0⟩↔|1⟩), 180° rotation around X-axis
- Y — bit-flip + phase-flip, 180° rotation around Y-axis
- Z — phase-flip (|1⟩→-|1⟩), 180° rotation around Z-axis
Key properties: Hermitian (→ valid observables) and unitary (→ valid gates). Each squares to I. They anticommute pairwise.
Why they matter: any single-qubit Hermitian matrix (observable/Hamiltonian term) is a weighted sum of I,X,Y,Z. Tensor products extend this to any number of qubits — the universal “alphabet” for quantum observables.
Related
- SparsePauliOp — how Pauli strings are represented in Qiskit
- X Gate
- Z Gate and Relative Phase
- Pauli Error Model — the same four operators used as a probabilistic noise model instead of deterministic gates
- Bloch Sphere — the “180° rotation around an axis” language above, made geometric
Self-Check
- Could you list all four Pauli operators and what each one does to a qubit?
- Why does “Hermitian and unitary” matter — what does each property buy you?
- Why can the same four operators describe both a deterministic gate and a probabilistic noise model?