The rules of calculation
Here you'll find everything the four locks of Box 2 ask for — no question needs knowledge from outside this page.
1 · Amplitudes and their square
A qubit is written as |ψ⟩ = α·|0⟩ + β·|1⟩. The two numbers α and β are called amplitudes. They are not probabilities — you only get those by squaring.
Since every measurement gives exactly one result, the two probabilities must add up to 1. This is the normalisation:
Calculating forwards
Given |ψ⟩ = 0.6·|0⟩ + 0.8·|1⟩:
- P(0) = 0.6² = 0.36 — that is 36 %
- P(1) = 0.8² = 0.64 — that is 64 %
- Check: 0.36 + 0.64 = 1 ✓
The most common trap: reading 0.8 as “80 %”. Only the square is the probability.
Calculating backwards
Given |ψ⟩ = 0.6·|0⟩ + b·|1⟩ with real b > 0. Find b:
- Set up the normalisation: 0.6² + b² = 1
- Rearrange: b² = 1 − 0.36 = 0.64
- Take the square root: b = 0.8
2 · Why 2n changes everything
A single qubit needs two amplitudes: one for |0⟩, one for |1⟩. Two qubits have four possible results — |00⟩, |01⟩, |10⟩, |11⟩ — and therefore need four amplitudes. With every additional qubit the number doubles.
| Qubits | Amplitudes | Order of magnitude |
|---|---|---|
| 1 | 2 | trivial |
| 10 | 1 024 | 2¹⁰ — fits into any pocket calculator |
| 20 | 1 048 576 | already over a million |
| 300 | 2³⁰⁰ | more than there are atoms in the observable universe |
That is exactly why a larger quantum computer can no longer be simulated on classical computers — not because it would be “faster”, but because the bookkeeping explodes.
3 · What a measurement does
Before the measurement a qubit is in superposition. The measurement gives exactly one result, and afterwards the state really is that result — the superposition is gone. This is called collapse.
Superposition is not ignorance
With a covered coin, the result has long been fixed; we just don't know it. With a qubit it is not fixed. The difference is measurable: only genuine superpositions can overlap and cancel each other out. Mere ignorance can never do that — where nothing oscillates, nothing cancels.
The test with the Hadamard gate is based on exactly this: applying H twice in a row brings |0⟩ back to exactly |0⟩. If the intermediate state had only been an unknown coin toss, 1 would come out in half of the cases.
No consciousness required
“Measuring” means an interaction that carries information to the outside — a detector, a photographic plate, a molecule of air. No person needs to look. A detector measures at night too, when nobody is in the lab. That large objects never appear in superposition has the same cause: they constantly interact with their surroundings — decoherence.
4 · Gates are rotations
Every qubit state is a point on the Bloch sphere. The angle θ measures how far the state is from the north pole |0⟩. A gate rotates this point — nothing more.
| θ | P(1) | State |
|---|---|---|
| 0° | 0 | |0⟩ — north pole |
| 60° | 0,25 | sin²(30°) = 0.25 |
| 90° | 0,5 | Equator, e.g. |+⟩ |
| 120° | 0,75 | sin²(60°) = 0.75 |
| 180° | 1 | |1⟩ — south pole |
A rotation Ry(φ) about the y-axis increases θ directly by φ. So if you start at |0⟩ (θ = 0°) and apply Ry(120°), you end up at θ = 120° — and therefore at P(1) = 0.75.
The four you need
5 · How much energy is in light?
Light comes in portions — photons. The energy of one portion depends only on the frequency, not on the brightness. Brighter light means more photons, not stronger ones.
In practice nobody calculates with these powers of ten. Measure energy in electronvolts and wavelength in nanometres, and the formula shrinks to a number you can remember:
Example
What is the energy of a photon with λ = 620 nm?
- Set up the formula: E = 1240 / 620
- Calculate: E = 2.0 eV
- Sanity check: 620 nm is orange-red — the low-energy end of visible light
| Wavelength | Energy | Colour |
|---|---|---|
| 700 nm | 1,8 eV | deep red |
| 620 nm | 2,0 eV | orange-red |
| 500 nm | 2,5 eV | green |
| 400 nm | 3,1 eV | violet |
Memory aid: short wavelength, high energy. That's why UV light gives you sunburn and red light doesn't.
6 · The Bell state
Two qubits can be linked in such a way that neither has a state of its own — only the pair together. The best-known such state:
The bracket contains only |00⟩ and |11⟩. Whatever isn't there has amplitude 0 — and by the Born rule therefore probability 0:
| Result | Amplitude | Probability |
|---|---|---|
| 00 | 1/√2 | 50 % |
| 01 | 0 | 0 % — never occurs |
| 10 | 0 | 0 % — never occurs |
| 11 | 1/√2 | 50 % |
So if you measure the first qubit and get 0, the second one gives 0 as well, with certainty — no matter how far apart the two are. Seen on its own, each of the two qubits is still completely random: 50 to 50.
What it is already used for today
The tangible use is eavesdropping-proof key exchange, quantum cryptography. Two parties share entangled pairs and obtain a common random sequence from them as a secret key. An eavesdropper would have to measure — and every measurement detectably destroys the state. So the attack gives itself away instead of going unnoticed. Such links are already running between cities and via satellite.
7 · What quantum technology can do — and what it can't
Quantum physics is not exotic
It doesn't only apply to the tiniest things in the lab. Lasers, LEDs, solar cells, the transistors in every phone chip, MRI in hospitals and the atomic clocks in GPS all rely on it. It is the best-confirmed theory in physics — and it's in your pocket.
A quantum computer does not try everything
The most widespread misconception. All possibilities are in play at the same time — but you may only measure once, and the answer would come out at random. It only becomes useful if you first cancel out the wrong possibilities through interference and reinforce the right one. That's exactly where most ideas for quantum algorithms fail.
And it won't replace your phone either
A quantum computer is a specialist tool, not a better all-rounder: superior at certain tasks, slower than your laptop at most others. For videos, chats and games, a normal computer stays clearly ahead. On top of that comes the effort — cooling almost to absolute zero and heavy shielding, because every disturbance destroys the superposition. The most promising use is simulating molecules and materials: they are quantum systems themselves.