Probability Amplitudes Without Scary Math
An amplitude is a number attached to an outcome. It can be negative, or even complex. Square its size and you get the outcome's chance. For a real state, all the chances add up to exactly 1. Squaring wipes out the sign, which is called phase. But later gates can still use it.
What is an amplitude, if not a probability?
Why care? Every prediction in this course comes from one small rule. Learn it here and you can predict any one-qubit histogram by hand.
A probability is a chance: a number between 0 and 1. An amplitude is the number a quantum state attaches to an outcome. It says how strongly the state leans toward that outcome. Unlike a chance, an amplitude may be negative or complex. One step links the two: chance = size of the amplitude, squared.
Think of a slider on a mixing desk that can go below zero. A slider at +3 and one at −3 are equally loud. They only differ in direction. (Unlike a real slider, the "loudness" here is the square of the setting.)
Worked example. The amplitude −1/√2 ≈ −0.7071 gives a chance of (−0.7071) × (−0.7071) = 0.5. The minus sign disappears from the chance. But it is not meaningless. It records phase. Phase is how this amplitude's sign compares with the others. Later gates can use it.
- How do you square a complex amplitude?
- How do you check and repair normalisation?
- Can you choose any probability you like?
- Run it: dial in a 75/25 splitINTERACTIVE
- Try this: swap the splitINTERACTIVE
- Where does the phase information go?
- Does amplitude 0.5 mean a 50% chance?
You’ve read the opening of chapter 3. Pro unlocks the other 7 sections — plus every chapter of every course, with circuits you can run right on the page. That’s $11.99 a month, about the price of a coffee, or $99.99 a year (save 30%). The first chapter of every course, and the whole math course, stay free.