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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. The Pauli X gate swaps them; the Pauli Z gate leaves their labels unchanged but changes the relative phase in a superposition. These are different operations, even though both are represented by half-turns on the sphere.
What the Bloch sphere represents
A qubit is a normalized superposition of two computational-basis states:
|ψ⟩ = α|0⟩ + β|1⟩, with |α|² + |β|² = 1.
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Here, α and β are complex amplitudes. The basis kets are conventionally written as column vectors, |0⟩ = (1, 0)ᵀ and |1⟩ = (0, 1)ᵀ; they are orthonormal. Measuring in this basis returns 0 with probability |α|² and 1 with probability |β|².
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The Bloch sphere is a geometric representation of a pure single-qubit state, not the qubit’s physical location. Ignoring an overall global phase—which does not change observable measurement probabilities—the state can be parameterized as:
|ψ⟩ = cos(θ/2)|0⟩ + e^(iφ)sin(θ/2)|1⟩.
Its Bloch vector has coordinates (sin θ cos φ, sin θ sin φ, cos θ). The polar angle θ determines the z-coordinate, and φ sets the direction around the equator. Note the half-angle in the amplitudes: the Bloch-vector coordinates use θ, not θ/2. The sphere is a single-qubit visualization; it does not represent a general multi-qubit state as a single point on an ordinary sphere. Microsoft Learn explains the qubit state-vector and Bloch-sphere model.
Where |0⟩ and |1⟩ sit
The computational basis is also called the Z basis. On the Bloch sphere, |0⟩ is at the north pole, in the +z direction, and |1⟩ is at the south pole, in the −z direction. The kets are quantum states, not ordinary three-dimensional coordinate directions.
The equator helps connect the sphere’s x-axis to another pair of states:
|+⟩ = (|0⟩ + |1⟩)/√2lies at+x.|−⟩ = (|0⟩ − |1⟩)/√2lies at−x.
These are equal-weight superpositions; their relative signs distinguish their positions. Microsoft Learn’s Dirac-notation guide describes the basis kets and these superposition states.
What Pauli X does
The Pauli X matrix is X = [[0, 1], [1, 0]]. Applying it swaps the basis states:
X|0⟩ = |1⟩X|1⟩ = |0⟩
For an arbitrary state, linearity gives X(α|0⟩ + β|1⟩) = β|0⟩ + α|1⟩. In the computational basis, X exchanges the two amplitudes, so it acts like a NOT on basis-state labels.
Geometrically, X is a 180° rotation about the x-axis: it leaves the x-coordinate unchanged and reverses the y- and z-coordinates. Thus it swaps the north and south poles. The states |+⟩ and |−⟩ are eigenstates of X, with eigenvalues +1 and −1, respectively; the −1 eigenvalue adds a global sign to that state. The X matrix and its state action can be checked directly.
What Pauli Z does
The Pauli Z matrix is Z = [[1, 0], [0, −1]]. It acts on the basis states as follows:
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Z|0⟩ = |0⟩Z|1⟩ = −|1⟩
For |ψ⟩ = α|0⟩ + β|1⟩, the result is Z|ψ⟩ = α|0⟩ − β|1⟩. Z changes the sign of the |1⟩ amplitude relative to the |0⟩ amplitude. That relative phase can affect later quantum operations, even though the probabilities from an immediate measurement in the computational basis remain |α|² for 0 and |β|² for 1.
For the isolated input |1⟩, the minus sign in Z|1⟩ = −|1⟩ is an overall phase on that state. It does not change the result of a Z-basis measurement. A superposition is different: changing the sign of one component relative to the other is not a global phase.
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1Repair Windows errors before they cause bigger problems2Scan for outdated or missing drivers - takes under a minute3Clear out junk files and repair common Windows errorsOn the sphere, Z is a 180° rotation about the z-axis. It leaves the z-coordinate—and therefore the two poles—in place, while reversing the x- and y-coordinates. It is not a computational-basis NOT gate. Z also exchanges |+⟩ and |−⟩. The matrix and Bloch-sphere rotation description make the distinction explicit.
X and Z compared
| Gate | Matrix | Action on |0⟩ and |1⟩ | Action on α|0⟩ + β|1⟩ | Bloch-sphere effect |
|---|---|---|---|---|
| X | [[0, 1], [1, 0]] |
Swaps the states: |0⟩ ↔ |1⟩ |
β|0⟩ + α|1⟩ |
180° rotation about x; x stays fixed, y and z reverse. |
| Z | [[1, 0], [0, −1]] |
|0⟩ stays; |1⟩ gains a minus sign. |
α|0⟩ − β|1⟩ |
180° rotation about z; z stays fixed, x and y reverse. |
The matrix is the clearest check on what a gate does to amplitudes. The sphere shows the same operation geometrically, but a rotation’s axis alone does not mean the gate swaps computational-basis labels.
How to read a gate diagram without mixing up bit and phase
- Locate
|0⟩and|1⟩at+zand−z, not along the x-axis. - For X, check whether the state moves to the opposite z pole: X swaps the computational-basis states.
- For Z, check whether a superposition’s phase changes: Z reverses the sign of its
|1⟩component while preserving the basis labels. - Do not infer a measurable change from a global sign on a basis state alone; a relative phase between components is the meaningful distinction.
For a deeper introduction to quantum computing, the Stanford Encyclopedia of Philosophy overview recommends Nielsen and Chuang (2010) as a detailed resource.
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