Curriculum / Quantum Foundations / The Bloch Sphere: A Map of the Qubit
The Bloch Sphere: A Map of the Qubit
Meet the geometric picture that turns every qubit state into a point on a sphere.
The Bloch Sphere: A Map of the Qubit
You have now met quite a cast of characters: |0⟩ and |1⟩, the superpositions |+⟩ and |−⟩, the phase-shifted states |i+⟩ and |i−⟩, and the gates X, H, Z, S, and T that move between them. Time to meet the map that holds them all: the Bloch sphere. Every possible state of a single qubit is exactly one point on the surface of this sphere.
Latitude: How Much |0⟩ vs |1⟩
Start at the poles. The north pole is |0⟩ and the south pole is |1⟩. Any other state lives somewhere in between, and its latitude tells you the balance: the closer the point is to the north pole, the more likely you are to measure 0.
A state exactly on the equator, like |+⟩, is a perfect 50/50 split. Formally, a state at polar angle θ (measured down from the north pole) gives 0 with probability cos²(θ/2) and 1 with probability sin²(θ/2). At the pole θ = 0 that is certainty; on the equator θ = 90° it is a coin flip.
Longitude: The Phase
Latitude alone cannot distinguish |+⟩ from |−⟩: both are 50/50. What separates them is the longitude, the angle around the equator, and that angle is precisely the relative phase you met in the phase gates lesson.
- •|+⟩ = (|0⟩ + |1⟩)/√2 points along +X (phase 0)
- •|i+⟩ = (|0⟩ + i|1⟩)/√2 points along +Y (phase 90°)
- •|−⟩ = (|0⟩ − |1⟩)/√2 points along −X (phase 180°)
- •|i−⟩ = (|0⟩ − i|1⟩)/√2 points along −Y (phase 270°)
Putting latitude and longitude together, every single-qubit state can be written as
with θ the latitude angle and φ the longitude. Two numbers, one sphere, every qubit state.
States on opposite ends of any diameter are orthogonal: |0⟩ vs |1⟩, |+⟩ vs |−⟩, |i+⟩ vs |i−⟩. Each such pair forms a measurement basis, which is why the three axes X, Y, and Z of the sphere are exactly the three bases you met in the measurement bases lesson.
Gates Move Points Around the Sphere
Here is the payoff of the geometric picture: every single-qubit gate is a rotation of the sphere.
- •X rotates 180° about the X axis: the poles swap, so |0⟩ becomes |1⟩. That is why X is the quantum NOT.
- •Z rotates 180° about the Z axis: the poles stay put, but |+⟩ swings around the equator to |−⟩. A pure phase move, invisible at the poles.
- •S and T are smaller Z-axis rotations: 90° and 45° of longitude.
- •H rotates 180° about the diagonal axis halfway between X and Z: it exchanges the Z axis and the X axis, which is exactly why it turns |0⟩ (a pole) into |+⟩ (the equator) and back.
Measurement on the Sphere
A standard measurement asks the qubit: north or south? The answer is random, weighted by latitude, and afterwards the state jumps to the pole that was announced. Measuring in the X or Y basis is the same story with the sphere tipped on its side: the state is projected onto the corresponding axis.
Where This Goes Next
The next lesson makes the rotation picture quantitative: RX, RY, and RZ gates that rotate by any angle you choose, with their matrices. And if you want the full tour, including composite rotations, global phase, and what lives inside the ball when noise enters the story, the Quantum Gates Deep Dive track opens with a dedicated Bloch sphere lesson.
How this lesson works
A guided reading lesson with interactive knowledge checks. Concepts are explained step by step with circuit diagrams and runnable examples, and you confirm understanding before moving on.
Part of: Quantum Foundations
Learn the basics: qubits, gates, superposition, and measurement.
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