Poles: |0⟩ and |1⟩

Every single-qubit pure state lives somewhere on the surface of a unit sphere called the Bloch sphere. The two computational basis states $|0\rangle$ and $|1\rangle$ are not scattered randomly across that surface — they occupy opposite ends of the vertical axis, the north pole and south pole.

The north and south poles

A general single-qubit state can be written as

where $\theta \in [0, \pi]$ is the polar angle from the $+z$ axis and $\varphi \in [0, 2\pi)$ is the azimuthal angle in the $xy$ plane. Setting $\theta = 0$ gives

so $|0\rangle$ sits at the north pole $(\theta = 0)$, regardless of $\varphi$. Setting $\theta = \pi$ gives

which equals $|1\rangle$ up to a physically irrelevant global phase, placing $|1\rangle$ at the south pole $(\theta = \pi)$.

The $+z$ axis therefore points from the south pole toward the north pole, and a measurement in the computational basis is a test of whether the Bloch vector points up ($|0\rangle$) or down ($|1\rangle$).

From north pole to south pole

The only way to move from one pole to the other while staying on the sphere is to rotate by $\pi$ radians (180°). The Pauli-X gate does exactly that — it rotates by $\pi$ around the $x$ axis, sending the north pole ($|0\rangle$) to the south pole ($|1\rangle$) and vice versa:

This is why $X$ is called the quantum NOT gate.

Try it

Flip the qubit to the south pole — prepare $|1\rangle$ starting from the default $|0\rangle$ state. The grader checks your full statevector, so leaving the circuit empty will not pass.

After running, the Bloch sphere in the visualizer should show a vector pointing straight down — the south pole, the home of $|1\rangle$.

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