Learn the math,
one signal at a time.

A qubit is not a vague idea — it is two numbers and one rule. Work through ten short lessons, each with its formula and a real calculation you can run in the console.

|0⟩

Read "ket zero": the outcome a qubit gives when it is definitely 0.

|ψ⟩

The state itself — a shorthand name for the pair of amplitudes.

α, β

The two amplitudes. Ordinary numbers, allowed to be negative.

|α|²

The chance of measuring 0. Squaring turns an amplitude into a probability.

H X Z Ry

Gates: instructions that rewrite the amplitudes. A circuit is read left to right.

θ

An angle in degrees. Ry(θ) tilts the state by that much.

shots

How many times the same measurement is repeated so the odds become visible.

≈

Expected, not guaranteed — randomness means one run is rarely exact.

1. 01 / LESSON

   What is a qubit?

   A classical bit is 0 or 1. A qubit holds two numbers called amplitudes, written α|0⟩ + β|1⟩. Both of them can be non-zero at the same time.

   |ψ⟩ = α|0⟩ + β|1⟩

   Watch forSquaring both amplitudes gives exactly 1. Every valid qubit obeys this.

   sqrt(0.5)^2 + sqrt(0.5)^2 = 1Run it
2. 02 / LESSON

   The probability rule

   Square an amplitude to get the chance of measuring that outcome. Do it to both amplitudes and the two chances always add up to 100%.

   P(0) = |α|², P(1) = |β|²

   Watch forAn amplitude of 0.707 means a 50% chance. The square is what you can actually predict.

   (1/sqrt(2))^2 = 0.5Run it
3. 03 / LESSON

   Hadamard (H)

   The gate that creates a superposition: it turns a definite 0 into an even mix of 0 and 1. Apply it a second time and the two halves cancel back to 0.

   H|0⟩ = (|0⟩ + |1⟩) / √2

   Watch forEach amplitude is about 0.707. Square it and you get a fair coin.

   1 / sqrt(2) = 0.707106781187Run it
4. 04 / LESSON

   Pauli X

   The quantum NOT. It swaps the two amplitudes, so a state that was certainly 0 becomes certainly 1.

   X|0⟩ = |1⟩

   Watch forThe flip moves the 1 from one amplitude to the other. The total never changes.

   0^2 + 1^2 = 1Run it
5. 05 / LESSON

   Pauli Z

   Z flips the sign of the |1⟩ amplitude. Nothing you measure directly changes, but that sign decides how the state interferes later.

   Z(α|0⟩ + β|1⟩) = α|0⟩ − β|1⟩

   Watch forA negative amplitude still squares to a positive chance. The sign only shows up when amplitudes meet.

   (-1 * 1/sqrt(2))^2 = 0.5Run it
6. 06 / LESSON

   Rotation Ry(θ)

   Instead of jumping between states, Ry tilts the state by an angle θ. The chance of measuring 0 follows a cosine squared.

   P(0) = cos²(θ / 2)

   Watch forA 60° tilt leans the state toward 0: three chances in four.

   cos(60 deg / 2)^2 = 0.75Run it
7. 07 / LESSON

   Measurement

   Looking collapses the qubit to a single answer, 0 or 1. Run the same state many times — the "shots" — and the counts settle around the probabilities.

   1024 shots × 0.5 ≈ 512

   Watch forProbability predicts the average, not the exact count of one run.

   1024 * 0.5 = 512Run it
8. 08 / LESSON

   The Bloch sphere

   Any qubit state is a point on a sphere: north pole is |0⟩, south pole is |1⟩, and the equator is a perfect superposition.

   θ = 2·acos(|α|)

   Watch for90° puts the point on the equator — the most uncertain state there is.

   2 * acos(1/sqrt(2)) * 180 / pi = 90Run it
9. 09 / LESSON

   Keeping it normalised

   Amplitudes have to form a unit length. If their squares do not add to 1, the numbers are not a qubit yet, so you rescale them.

   |α|² + |β|² = 1

   Watch for0.6 and 0.8 look random, but their squares add to exactly 1.

   sqrt(0.6^2 + 0.8^2) = 1Run it
10. 10 / LESSON

    Interference

    Amplitudes are added first and squared afterwards, so two paths can cancel each other out or build each other up. That is what separates a quantum calculation from simply adding odds.

    (α₁ + α₂)²

    Watch forTwo equal amplitudes with opposite signs cancel to nothing — the reason H twice returns |0⟩.

    (1/sqrt(2) - 1/sqrt(2))^2 = 0Run it

Qubit

A two-outcome system described by two amplitudes. Measured, it gives 0 or 1.

Amplitude

A number attached to an outcome. Squared, it becomes the chance of that outcome.

Phase

The sign of an amplitude. Invisible in a single chance, decisive in interference.

Superposition

Both amplitudes non-zero at once, so neither outcome is settled before you look.

Gate

A rule that rewrites the amplitudes. H, X, Z and Ry are the ones used here.

Collapse

What measurement does: the state reduces to the single outcome you saw.

Shot

One repetition of the same measurement. Many shots reveal the probabilities.

Normalise

Rescale the amplitudes so their squares add to 1.

Interference

Amplitudes combining — reinforcing or cancelling — before you square them.

Bloch sphere

A picture of every possible single-qubit state as a point on a sphere.