Visualize any single-qubit state as a point on a 3D sphere
Every state of a single qubit is a point on a sphere, and every gate is a rotation of it. Spin it, push the state around, and watch.
Drag to orbit the sphere and use the sliders to move the glowing state. |0⟩ is the north pole, |1⟩ the south pole, and the equator is a 50/50 split.
The Bloch sphere is a beautiful geometric representation of a single qubit. Every pure state |ψ⟩ = cos(θ/2)|0⟩ + eiφsin(θ/2)|1⟩ maps to a unique point on the surface of a unit sphere, where θ is the polar angle and φ is the azimuthal angle.
|0⟩ sits at the north pole, |1⟩ at the south pole, and every other state lives somewhere on the surface. Quantum gates are simply rotations of this sphere.
Drag to orbit the sphere. Use the sliders to move the state vector (the glowing point) anywhere on the surface. Watch how the state equation, probabilities, and Bloch coordinates update in real time.
North Pole: θ = 0 → |0⟩ with 100% probability of measuring 0
South Pole: θ = π → |1⟩ with 100% probability of measuring 1
Equator: θ = π/2 → Equal superposition, 50/50 outcomes. Phase φ varies around the equator
Pick a gate and click "Apply Gate" to watch the state rotate. X flips |0⟩ to |1⟩; H turns |0⟩ into |+⟩ on the equator.
Every single-qubit gate corresponds to a rotation of the Bloch sphere. The X gate (NOT) is a 180° rotation around the X-axis — it flips |0⟩ to |1⟩. The Hadamard gate rotates 180° around an axis between X and Z, mapping |0⟩ to |+⟩. Click gates to see the state vector rotate in real time.
Gate Quick Reference
Apply gates and each rotation leaves a colored arc. Try the HZH = X preset: three arcs land exactly where a single X gate would.
Apply sequences of gates and watch the state trace colored arcs on the Bloch sphere. Each gate's rotation leaves a visible trail, building up a geometric record of the computation. Try the preset sequences to see famous gate identities visualized as paths that return to the same point.
Try this: Apply H → Z → H and watch it trace three arcs that end up at the same place as a single X gate. This is the identity HZH = X, visualized geometrically. The Hadamard gate "conjugates" Z into X by changing the rotation axis.
Click each of the six states to highlight it on the sphere. They sit in pairs at the ends of the X, Y and Z axes.
Six special states sit at the endpoints of the three axes. They form three mutually unbiased bases — measuring a state from one basis gives completely random results in the other two bases. Click each state to highlight it on the sphere.
Three bases — Z (computational), X (Hadamard), and Y (circular) bases correspond to the three sphere axes
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