The Pauli operators (or Pauli matrices) are the three fundamental single-qubit gates: X (bit flip, |0⟩↔|1⟩, like classical NOT), Y (combined bit and phase flip), and Z (phase flip, |0⟩→|0⟩, |1⟩→−|1⟩). Together with the identity I, they form a complete basis for all 2×2 Hermitian matrices and are foundational to quantum computing. The X gate is the quantum NOT gate. The Z gate introduces a relative phase of −1 to the |1⟩ state. The Y gate = iXZ. Pauli operators are Hermitian and unitary: P† = P and P² = I (applying any Pauli twice returns to the original state). They are used extensively in quantum error correction (Pauli errors), Hamiltonian simulation (Pauli decomposition of Hamiltonians), and variational algorithms (Pauli expectation values in VQE).
Related Terms
Quantum Gate
GatesA unitary operation that transforms the state of one or more qubits.
Hadamard Gate
GatesThe H gate — creates an equal superposition of |0⟩ and |1⟩ from a basis state.
CNOT Gate
GatesControlled-NOT — a two-qubit gate that flips the target qubit when the control qubit is |1⟩.
Quantum Error Correction
HardwareTechniques to detect and correct errors in quantum circuits without measuring (and collapsing) the qubits.