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Use Pauli X to swap the computational-basis states |0⟩ and |1⟩. Use Pauli Z to leave those labels unchanged while changing the relative phase of the |1⟩ amplitude. The right gate depends on the state transformation you need—or, in error-correction language, whether you are addressing a bit flip or a phase flip.
What Pauli X and Pauli Z do
The Pauli gates are single-qubit unitary operations. For a general qubit α|0⟩ + β|1⟩, where α and β are complex amplitudes, their actions are:
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| Property | Pauli X | Pauli Z |
|---|---|---|
| Matrix | [[0, 1], [1, 0]] | [[1, 0], [0, −1]] |
| On computational-basis states | X|0⟩ = |1⟩; X|1⟩ = |0⟩ | Z|0⟩ = |0⟩; Z|1⟩ = −|1⟩ |
| On α|0⟩ + β|1⟩ | β|0⟩ + α|1⟩ | α|0⟩ − β|1⟩ |
| Common description | Bit flip or NOT-like operation | Phase flip |
| Bloch-sphere interpretation | π rotation about the x axis | π rotation about the z axis |
| Error-correction shorthand | Bit-flip error | Phase-flip error |
IBM Quantum Learning calls X a bit flip or NOT operation and Z a phase flip in its Single systems lesson. The names describe different effects: X exchanges basis-state labels, while Z adds a minus sign to the |1⟩ component.
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Choose X when the intended operation is to exchange |0⟩ and |1⟩. On a computational-basis input, X|0⟩ = |1⟩ and X|1⟩ = |0⟩. On a superposition, it exchanges the amplitudes: α|0⟩ + β|1⟩ becomes β|0⟩ + α|1⟩.
On the Bloch sphere, X is a π rotation about the x axis. The plus/minus states, |+⟩ = (|0⟩ + |1⟩)/√2 and |−⟩ = (|0⟩ − |1⟩)/√2, are eigenstates of X, so applying X to either leaves the physical state unchanged apart from a possible overall sign.
When to use Z
Choose Z when the basis labels should stay put but the |1⟩ amplitude should acquire a minus sign. It maps α|0⟩ + β|1⟩ to α|0⟩ − β|1⟩. On |0⟩, Z has no effect; on |1⟩, it returns −|1⟩.
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That sign is a relative phase when both basis components are present, and relative phase can change what happens in later parts of a circuit. For example, Z turns |+⟩ into |−⟩. These states have the same probabilities when measured directly in the computational basis, but a Hadamard gate distinguishes them: H|+⟩ = |0⟩, whereas H|−⟩ = |1⟩.
On the Bloch sphere, Z is a π rotation about the z axis. The computational basis is associated with that axis, which helps explain why Z preserves the basis labels while changing the phase between them.
Choosing a gate for an error or transformation
- Need to exchange |0⟩ and |1⟩? Use X; its error-correction counterpart is a bit-flip error.
- Need to change the relative phase between |0⟩ and |1⟩? Use Z; its error-correction counterpart is a phase-flip error.
- Need a specific operation on a superposition? Apply each gate to the amplitudes and check the resulting state, rather than judging by basis-state probabilities alone.
In stabilizer and error-correction notation, these are distinct Pauli error types, not interchangeable names for the same effect. The Pauli Y operation is equivalent to XZ up to a phase, and X and Z anticommute: XZ = −ZX. Each of X and Z is its own inverse, so XX = ZZ = I. IBM Quantum Learning discusses these relations in its stabilizer-formalism lesson.
Pauli gates and parameterized rotations are related, but not identical
A π rotation about an axis is closely related to the Pauli gate for that axis, but the matrices differ by a global phase: RX(π) = −iX and RZ(π) = −iZ, as documented in the Qiskit XGate and ZGate API entries. For an isolated state, an overall global phase has no observable effect. In controlled constructions, however, phase bookkeeping can matter, so do not substitute one expression for the other without considering the circuit context.
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The symbols X and Z can refer to gates, observables, or error operators, depending on context. The matrix actions above describe the gates. In error-correction discussions, “X error” and “Z error” identify the corresponding bit-flip and phase-flip error types. Check how the notation is being used before interpreting a symbol as an instruction to apply a gate.
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