Classical computing versus quantum computing

A classical bit has one of two values: 0 or 1. A pure qubit is described by two complex amplitudes. Those amplitudes let us predict measurement outcomes and, through their relative phase, how later operations will interfere.

What changes?

Write a qubit as

$$\lvert\psi\rangle=\alpha\lvert0\rangle+\beta\lvert1\rangle,\qquad |\alpha|^2+|\beta|^2=1.$$

You do not need to know complex numbers yet. For now, read this as: “a state is built from two amplitudes, and the probabilities must add to one.” Measuring in the computational, or Z, basis gives one bit: 0 with probability $|\alpha|^2$, or 1 with probability $|\beta|^2$.

A qubit does not give you two independently readable classical values. The state is richer than the outcome of one measurement.

A first experiment

The sphere below represents a single pure qubit. Its north pole is $\lvert0\rangle$ and south pole is $\lvert1\rangle$. Start at the north pole, apply H, then run repeated measurements. Every shot prepares the same state again.

L01 · INTERACTIVE LABRuns on your device

Qubit explorer

A pure qubit is a normalised pair of complex amplitudes. The sphere shows the state; the bars show measurement probabilities in the Z basis.

Make a prediction first

Move the state to the equator. What will happen to the Z-basis probabilities when you change only the phase?

Experiment controls

Polar angle θ90°Relative phase φ0°

HXYZST

Prepare |0⟩Random seed

Run measurements →Reset

Exact probability

|0⟩

50.0%

|1⟩

50.0%

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

Create a state with a 75% probability of measuring 1. Can you find two different phases with the same probabilities?

Prediction: Move the state to the equator. What will happen to the Z-basis probabilities when you change only the phase? Challenge: Create a state with a 75% probability of measuring 1. Can you find two different phases with the same probabilities?

A different workflow

1. Prepare the input state.
2. Transform it using gates.
3. Measure to obtain classical outcomes.
4. Analyse repeated outcomes and compare them with a prediction.

Classical computers still handle the surrounding application: data, user interfaces, networking and analysis. Quantum methods may help with particular structured problems. A large search space by itself does not prove that a quantum approach will be practically faster.

PAUSE & THINK

What do you get from one Z-basis measurement of a qubit?

Choose your answer

Both amplitudes, read exactlyOne classical outcome: 0 or 1Every possible solution to a problem

Check my answer

Connect it to the source

This lesson develops sections 2–4 of the author’s original introduction. Continue to quantum information or take the gentle route to qubits.

Reference: IBM Quantum — Basics of quantum information.