Chapter 02 / 20

Complex numbers and probability amplitudes

Magnitude, phase, normalization, and interference.

1. Core idea

A complex amplitude has both magnitude and phase. The magnitude squared determines a measurement probability, while relative phases control what happens when paths interfere. A common global phase has no measurable effect on an isolated state.

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

The states (|0⟩+|1⟩)/√2 and (|0⟩−|1⟩)/√2 have identical computational-basis measurement probabilities but respond differently to a Hadamard gate.

4. Reasoning flow

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

Hands-on activity

Use a two- or three-qubit circuit to model the stated example. The worked calculation above gives a reference against which to check the simulated output.

5. Check your understanding

Practice question

Calculate probabilities for amplitudes 3/5 and 4i/5, then check normalization.

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.

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✦Complex numbers and probability amplitudes