✦Single-qubit states and the Bloch sphere

Geometric understanding of two-level systems.

1. Core idea

Ignoring global phase, every pure qubit corresponds to a point on a sphere. Polar angle θ controls the measurement probabilities and azimuth φ controls the relative phase. Mixed states occupy the interior of the Bloch ball.

2. Mathematical foundation

The notation describes a precise relationship among states, transformations or probabilities. Identify each symbol, check the dimensions of the objects, and distinguish amplitudes from the probabilities produced by a measurement.

3. Worked example

4. Reasoning flow

1. Define the initial register and its basis states.
2. Apply the transformations in their intended order.
3. Track amplitude magnitudes and relative phases.
4. Measure in the basis that reveals the desired property.
5. Interpret the result and check the resource assumptions.

Hands-on activity

5. Check your understanding

6. What to watch out for

Ideal-state calculations assume perfect gates and measurements. When assessing an algorithm, separate its mathematical guarantee from the cost of building oracles, preparing inputs, repeating experiments and correcting physical noise.