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On a Bloch sphere, the computational-basis states |0⟩ and |1⟩ sit at the north and south poles of the z-axis. Pauli X swaps those states; Pauli Z leaves their labels unchanged but changes the relative phase in a superposition. The distinction is the key to reading the diagram without mistaking a phase change for a bit flip.
What the Bloch sphere represents
A qubit is described by two complex amplitudes:
|ψ⟩ = α|0⟩ + β|1⟩, where |α|² + |β|² = 1.
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The basis kets are conventionally written as column vectors |0⟩ = (1, 0)ᵀ and |1⟩ = (0, 1)ᵀ. They are orthonormal. If the qubit is measured in this computational (Z) basis, the outcome is 0 with probability |α|² and 1 with probability |β|². Stanford Encyclopedia of Philosophy: Quantum Computing
The Bloch sphere is a geometric representation of a pure single-qubit state, not the physical location of a qubit. It also does not represent an arbitrary multi-qubit state as one point on an ordinary sphere. Microsoft Learn: The Qubit in Quantum Computing
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Where |0⟩ and |1⟩ sit
Ignoring an overall global phase, a pure qubit can be parameterized as:
|ψ⟩ = cos(θ/2)|0⟩ + eiφsin(θ/2)|1⟩.
Its Bloch vector is (sin θ cos φ, sin θ sin φ, cos θ). The half-angle in the amplitudes matters: the vector uses θ, while the state coefficients use θ/2. The polar angle θ determines the z-coordinate, and φ sets the direction around the equator. Quantum Education Modules: Bloch Sphere
- |0⟩: north pole, +z.
- |1⟩: south pole, −z.
- |+⟩ = (|0⟩ + |1⟩)/√2: +x point on the equator.
- |−⟩ = (|0⟩ − |1⟩)/√2: −x point on the equator.
The labels |0⟩ and |1⟩ are kets, not ordinary Cartesian direction vectors. Their locations on this representation are the two ends of the z-axis. Microsoft Learn: Dirac Notation in Quantum Computing
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What Pauli X does
The Pauli X matrix is:
X = [[0, 1], [1, 0]].
Applied to the computational-basis states, it swaps them: X|0⟩ = |1⟩ and X|1⟩ = |0⟩. On a Bloch vector, X is a 180° rotation about the x-axis. This leaves the x-coordinate unchanged and reverses the y- and z-coordinates. Microsoft Learn: The Qubit in Quantum Computing
For an arbitrary state, X(α|0⟩ + β|1⟩) = β|0⟩ + α|1⟩. It exchanges the amplitudes rather than acting as a phase-only change in the computational basis.
What Pauli Z does—and why it is not a bit flip
The Pauli Z matrix is:
Z = [[1, 0], [0, −1]].
It acts on the basis states as Z|0⟩ = |0⟩ and Z|1⟩ = −|1⟩. For a single basis-state input, the minus sign is an overall phase and does not change the result of a Z-basis measurement. The labels 0 and 1 are not swapped.
For a superposition, Z(α|0⟩ + β|1⟩) = α|0⟩ − β|1⟩. The sign change is now relative between the two amplitudes, so it can affect later interference and measurements in other bases. Geometrically, Z is a 180° rotation about z: it leaves the z-coordinate unchanged and reverses x and y. Microsoft Learn: The Qubit in Quantum Computing
Thus X is the computational-basis NOT operation; Z is a phase operation in that basis. Calling both gates “bit flips” hides their different effects.
Compare X and Z at a glance
| Gate | Matrix | Action on |0⟩ and |1⟩ | Action on an arbitrary state | Bloch-sphere effect |
|---|---|---|---|---|
| X | [[0, 1], [1, 0]] | Swaps |0⟩ and |1⟩ | α|0⟩ + β|1⟩ → β|0⟩ + α|1⟩ | 180° rotation about x |
| Z | [[1, 0], [0, −1]] | |0⟩ stays |0⟩; |1⟩ becomes −|1⟩ | α|0⟩ + β|1⟩ → α|0⟩ − β|1⟩ | 180° rotation about z |
The matrix forms and rotation descriptions are consistent ways to check the same operation. Introduction to Quantum Information Science: 2.10 The Bloch sphere
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Use |+⟩ and |−⟩ as a cross-check
The equatorial x-axis states make the gate geometry easier to verify. X|+⟩ = |+⟩ and X|−⟩ = −|−⟩, so they are X eigenstates with eigenvalues +1 and −1. Z swaps them: Z|+⟩ = |−⟩ and Z|−⟩ = |+⟩. The minus sign in X|−⟩ is an overall phase on that state, not a different physical state.
A quick way to read a diagram
- Locate the z-axis poles first: |0⟩ is +z and |1⟩ is −z.
- Check whether the operation swaps pole states or leaves their labels in place. X swaps them; Z does not.
- For a superposition, track both amplitudes and their relative sign or phase. Z changes the relative phase even though it leaves each basis label intact.
- Interpret the gate as a half-turn about its named axis: X about x, Z about z.
These checks prevent the common errors of treating basis kets as Cartesian axes, omitting θ/2 from the state parameterization, or concluding that Z|1⟩ changes a Z-basis measurement outcome.
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For a fuller treatment of quantum computing, the Stanford Encyclopedia of Philosophy’s overview recommends Nielsen and Chuang (2010) as a detailed introduction. Stanford Encyclopedia of Philosophy: Quantum Computing
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