Superposition

A useful starting state

Apply H to $\lvert0\rangle$:

$$H\lvert0\rangle=\lvert+\rangle=\frac{\lvert0\rangle+\lvert1\rangle}{\sqrt2}.$$

Both Z outcomes have probability $1/2$. The state is a coherent combination of amplitudes relative to this basis. Superposition is basis-dependent: $\lvert+\rangle$ is itself one basis vector in the X basis.

Why a coin is an incomplete analogy

A classical random bit can also produce equal 0/1 counts. Those counts alone do not distinguish it from $\lvert+\rangle$. The difference appears when you apply another operation.

Applying H again maps $\lvert+\rangle$ to $\lvert0\rangle$ deterministically. An equal classical mixture of $\lvert0\rangle$ and $\lvert1\rangle$ stays 50/50 under H. The coherent state has a stable relative phase; the mixture does not.

Turn a phase into an outcome

Insert Z between the two H gates. It changes $\lvert+\rangle$ into $\lvert-\rangle$. The second H now gives $\lvert1\rangle$. The intermediate Z-basis probabilities did not change, but the final outcome did.

L03 · INTERACTIVE LABRuns on your device

Superposition & interference

The two Hadamard gates recombine amplitudes. A relative phase between them changes the result even though the intermediate Z-basis probabilities stay at 50/50.

Make a prediction first

H then H returns 0. What do you expect if a Z gate is placed between them?

Experiment controls

|0⟩ — H — P(φ) — H — M

Relative phase φ0°

H · HH · Z · H

P(φ) = diag(1, e^(iφ)). P(180°) is the Z gate.

Random seed

Run measurements →Reset

Exact probability

|0⟩

100.0%

|1⟩

0.0%

Exact ideal-model predictions · run a measurement experiment to compare.

MODEL · Ideal pure-state vector, complex double precisionORDER · q0 is the leftmost bit

Take it further

Set the phase to 180°. Explain why the final probability of 1 becomes 100%.

Prediction: H then H returns 0. What do you expect if a Z gate is placed between them? Challenge: Set the phase to 180°. Explain why the final probability of 1 becomes 100%.

Practise

Compare phase settings 0°, 90° and 180°. Calculate or read the final probability of 1 for each. You should find 0, $1/2$ and 1. Explain the result using amplitudes rather than a claim that measurement reads both possibilities.

Superposition gives algorithms a state space to transform. It does not by itself reveal exponentially many answers, eliminate data-loading costs or guarantee a speedup.

Reference: IBM Quantum — Quantum mechanics basics.