Every state of a single qubit is a point on a sphere. The Bloch sphere turns abstract quantum amplitudes into a picture you can rotate and reason about.
Bloch SphereLive
the state of a qubit
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Every pure state of a single qubit is a point on the Bloch sphere. The north pole is |0⟩, the south |1⟩, and the equator holds equal superpositions differing only in phase φ. Quantum gates rotate this arrow — the geometric picture behind all single-qubit quantum computing.
Reading this result: Polar angle θ sets the measurement odds: P(|0⟩)=cos²(θ/2)=0.75, so tilting toward a pole biases the qubit while φ only twists its phase.
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How it works
A qubit state cos(θ/2)|0⟩ + e^{iφ}sin(θ/2)|1⟩ maps onto a unit sphere: the poles are |0⟩ and |1⟩, the equator holds equal superpositions differing only by phase. Measurement probabilities come straight from the polar angle. Quantum gates are rotations of this arrow — the geometric heart of quantum computing.
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The math, the assumptions, real-world uses, or a code translation — explained for this exact simulation.
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The solver uses established numerical methods, but results are for research and educational purposes and should be validated against experiment or professional review before you rely on them.