03 · Amplitudes
Interference
Probabilities add for coins. Amplitudes add for qubits — and a minus sign can cancel a path. That is the whole advantage.
In English
Coin probabilities only add. Quantum amplitudes can cancel: a minus sign on one path can kill an outcome that both paths would have allowed. That cancellation is the whole reason a quantum algorithm can be faster.
- Try this
- Start from |+⟩. Apply Z (a phase). Then Hadamard. Watch a path disappear that was there before the minus.
- Keep this
- Amplitudes add, and a minus can cancel a path. Probabilities cannot do that.
Lab · H — phase — H
A qubit analogue of a Mach–Zehnder interferometer. Two Hadamards are the beam splitters. The phase is a path length. If you ask which path, the second splitter has nothing left to cancel.
- 1. H
- 2. P(φ)
- 3. H
Closed P(|0⟩) = cos²(φ/2) = 100.0%
|0⟩1100%|1⟩00%
Two classical paths: add the probabilities. Two quantum paths: add the amplitudes, then square. If they point opposite ways, the total can be zero. A relative minus sign is a physical resource.
Distinguish the paths — measure which one — and you no longer add amplitudes. The pattern dies. That is why a quantum algorithm is an interference experiment, not a pile of parallel coins.