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Pauli X flips the computational-basis states and rotates a qubit 180° around the Bloch sphere’s x axis. Pauli Z leaves the basis labels in place, changes the sign of the |1⟩ amplitude, and rotates 180° around the z axis. For a general superposition, X exchanges the two amplitudes; Z changes their relative sign. That distinction is why X is called a bit flip and Z a phase flip.
Start with a general qubit
A pure qubit can be written as |ψ⟩ = α|0⟩ + β|1⟩, where α and β are complex amplitudes and |α|² + |β|² = 1. The Pauli gates are represented by these matrices:
| Gate | Matrix | Effect on |ψ⟩ |
|---|---|---|
| X | [[0, 1], [1, 0]] | β|0⟩ + α|1⟩ |
| Z | [[1, 0], [0, −1]] | α|0⟩ − β|1⟩ |
The formulas show the essential difference: X swaps which basis state carries each amplitude, while Z preserves the basis labels and negates the |1⟩ amplitude. The matrices and gate descriptions are documented in IBM Quantum Learning’s “Bits, gates, and circuits” and the Qiskit ZGate reference.
What does the X gate do to a qubit?
Applying X gives X|ψ⟩ = β|0⟩ + α|1⟩. On the computational-basis states, it swaps |0⟩ and |1⟩:
- X|0⟩ = |1⟩
- X|1⟩ = |0⟩
This basis-state action is analogous to a classical NOT operation, so X is often called a bit flip. On the Bloch sphere, X is a π-radian (180°) rotation around the x axis. The corresponding Bloch-vector transformation is (x, y, z) → (x, −y, −z): the x coordinate stays fixed, while y and z reverse sign.
“Bit flip” is shorthand for what X does to the computational-basis states, not a claim that every superposition becomes a classical bit. For an arbitrary qubit, X exchanges α and β, and the result may still be a superposition.
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What does the Z gate do to a qubit?
Applying Z gives Z|ψ⟩ = α|0⟩ − β|1⟩. It leaves |0⟩ unchanged and multiplies |1⟩ by −1:
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- Z|1⟩ = −|1⟩
This is called a phase flip: for a superposition, the sign change alters the relative phase between the |0⟩ and |1⟩ components rather than swapping them. Z is a π-radian (180°) rotation around the z axis, mapping a Bloch vector (x, y, z) to (−x, −y, z). The z coordinate remains fixed; x and y reverse sign. See the Qiskit ZGate reference for the matrix, phase-flip description, and rotation convention.
How the rotations appear on the Bloch sphere
A Bloch-sphere point represents a pure-qubit state up to an overall, or global, phase. The computational basis is aligned with the z axis: |0⟩ is the north pole and |1⟩ is the south pole. Consequently, a half-turn around x exchanges the poles, while a half-turn around z leaves both pole locations fixed.
That does not mean Z has no effect on the state vector |1⟩: it changes |1⟩ to −|1⟩. For |1⟩ alone, this is only a global phase and does not change its physical state. When |0⟩ and |1⟩ amplitudes coexist, however, changing the sign of just one component changes the relative phase and can change later interference or measurement outcomes. IBM’s lesson describes measurement along z and uses |+⟩ as an equal superposition: IBM Quantum Learning, “Lesson 02: Bits, gates, and circuits”.
A quick example: X and Z on |+⟩
Let |+⟩ = (|0⟩ + |1⟩)/√2, an equal superposition. Then:
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- X|+⟩ = |+⟩: exchanging two equal amplitudes leaves the state unchanged.
- Z|+⟩ = |−⟩ = (|0⟩ − |1⟩)/√2: the |1⟩ amplitude changes sign.
|+⟩ and |−⟩ have the same probabilities when measured in the computational basis, but their relative phases differ. The states occupy opposite points on the Bloch sphere’s equator, illustrating how Z can change a qubit without changing its immediate computational-basis probabilities.
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X versus Z at a glance
| Property | Pauli X | Pauli Z |
|---|---|---|
| Matrix | [[0, 1], [1, 0]] | [[1, 0], [0, −1]] |
| Basis-state action | |0⟩ ↔ |1⟩ | |0⟩ → |0⟩; |1⟩ → −|1⟩ |
| Bloch-sphere rotation | 180° around x | 180° around z |
| Bloch-vector map | (x, y, z) → (x, −y, −z) | (x, y, z) → (−x, −y, z) |
| Common name | Bit flip | Phase flip |
Global phase and rotation-gate conventions
Rotation-gate formulas can differ from Pauli matrices by an overall phase. Under the convention documented by Qiskit, RZ(π) = −iZ, not literally Z as a matrix; the factor −i is a global phase and does not change the physical action on an isolated qubit. Qiskit notes the same kind of convention for RX(π) = −iX. Compare physical states up to global phase rather than assuming every named rotation matrix exactly equals its Pauli gate.
Further reading
For a broader treatment of quantum gates and quantum circuits, Cambridge University Press lists Nielsen and Chuang’s Quantum Computation and Quantum Information, 10th Anniversary Edition, published in 2010.
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