The Bloch sphere

The Bloch sphere is the most useful picture in all of quantum computing. Every possible state of a single qubit — every superposition, every phase, every weird mix — corresponds to exactly one point on its surface. One-to-one. No exceptions.

This lesson is mostly a playground. Drag the sliders, jump between named states, watch the state vector and the amplitude clocks update together. Your job is to build a physical intuition for what a qubit is.

Anatomy of the sphere

The Bloch sphere is a unit sphere (radius $1$) centered at the origin, with three landmarks:

• The north pole is $|0\rangle$. $\theta = 0$.
• The south pole is $|1\rangle$. $\theta = \pi$.
• The equator is the set of states where $|0\rangle$ and $|1\rangle$ are mixed $50/50$. Which point on the equator you’re at is set by the phase $\varphi$.

Six states have special names because they keep showing up:

Click the state buttons in the explorer above to jump between them. Watch how the state vector in the top-right display changes. In particular, notice what happens to the phase of the amplitudes — the little clock hands — as you move around the equator while keeping $\theta = \pi/2$ fixed. That’s the phase $\varphi$ doing its work.

The formula

The two angles $\theta$ and $\varphi$ map to the qubit state via:

$|\psi\rangle = \cos\!\frac{\theta}{2}\,|0\rangle + e^{i\varphi}\sin\!\frac{\theta}{2}\,|1\rangle$

Don’t panic. You don’t need to memorize this. Just notice two things:

1. When $\theta = 0$: $\cos(0) = 1$, $\sin(0) = 0$, so $|\psi\rangle = |0\rangle$. ✓
2. When $\theta = \pi$: $\cos(\pi/2) = 0$, $\sin(\pi/2) = 1$, so $|\psi\rangle = e^{i\varphi}|1\rangle$. The phase out front is a global phase — it doesn’t change anything you can measure, so we usually drop it and just write $|1\rangle$. ✓

The half-angle $\theta/2$ in the formula is there because of a geometric quirk: two qubit states that are “opposite” on the Bloch sphere (like $|0\rangle$ and $|1\rangle$, at angle $\pi$ on the sphere) need to be fully orthogonal as quantum states (angle $\pi/2$ in state space). Halving $\theta$ makes the math work out.

Global vs relative phase — the subtle bit

Two qubit states that differ only by an overall complex factor like $e^{i\varphi}$ are physically the same state. You can’t tell them apart by any measurement. That’s called a global phase, and we throw it away.

But the relative phase between the $|0\rangle$ and $|1\rangle$ components — the $e^{i\varphi}$ in the formula above — is real and matters. It’s what makes $|+\rangle$ and $|-\rangle$ different. It’s what makes quantum interference possible. We’ll see it in action when we meet the Hadamard gate.

The takeaway

Every single-qubit state is a point on the Bloch sphere. Two angles, $\theta$ and $\varphi$, are enough to describe any of them. The picture is literal: if two states live at the same point on the sphere, they’re the same state.

Next up: what actually happens when you look at a qubit.