Bloch Sphere Simulator

The Bloch sphere is the standard way to picture the state of a single qubit. Any pure state can be written as

$$\lvert \psi \rangle = \cos\tfrac{\theta}{2}\,\lvert 0 \rangle + e^{i\varphi}\sin\tfrac{\theta}{2}\,\lvert 1 \rangle$$

and mapped to a point on the unit sphere with coordinates $(\sin\theta\cos\varphi,\ \sin\theta\sin\varphi,\ \cos\theta)$.

• North pole ($z=+1$) is $\lvert 0 \rangle$, south pole is $\lvert 1 \rangle$.
• The equator holds the equal-superposition states like $\lvert + \rangle = (\lvert 0\rangle + \lvert 1\rangle)/\sqrt{2}$.

Click the gate buttons below to apply gates and watch the state vector move.

Things to try

1. Apply H to $\lvert 0 \rangle$ — the vector swings to the $+x$ axis (the $\lvert + \rangle$ state).
2. From there, apply Z — it rotates to $-x$ (the $\lvert - \rangle$ state).
3. Apply H then S then H — this is a rotation you'll meet again in phase-estimation circuits.
4. Apply the same gate twice: X·X, H·H, Z·Z all return you to where you started (these gates are their own inverse).

Why it matters

Single-qubit gates are rotations of the Bloch sphere. Getting an intuition for how H, X, Y, Z, S and T move the state vector is one of the fastest ways to build real quantum intuition. When you're ready, combine qubits in the Circuit Builder and study the gate matrices in the Gate Visualizer.

See also the Beginner Path → Qubits & Gates.