You do not need this track to understand quantum computing. You need it if you want to see the symbols as well as the pictures. Start here.
A complex number sounds forbidding. It is a number with a length and a direction, like an arrow drawn on paper.
Two ways to write the same arrow
By its corners: z = a + bi. Go a steps right, then b steps up. The i just means "this bit goes up, not right". Handy for adding arrows.
By length and angle: z = r·e^(iθ). The arrow is r long and points at angle θ. Handy for turning things, because to turn an arrow you just add to the angle.
Quantum computing lives in the second form. A gate that "adds a phase" is literally just adding to θ.
The two things you will do with them
Take the length. |a + bi| = √(a² + b²) — plain Pythagoras. Square that length and you have a probability.
Flip it. (a + bi)* = a − bi, which mirrors the arrow across the horizontal. It is called the conjugate, and it turns up whenever you compare two arrows.
Phase
The angle part. It is invisible when you look at a qubit — but it decides whether two routes add up or cancel out. Every interesting thing in this field is a phase doing its job.
Why the minus sign is the whole point
Two routes arriving as +½ and −½ add up to nothing. Separately, each one would have given a one-in-four chance. Together: zero.
Ordinary probabilities cannot do that. They are never negative, so they only ever pile up. Cancelling needs a number system with a minus sign in it, and that is the only reason quantum mechanics insists on these.
Worth remembering
- A complex number is an arrow: a length and an angle.
- The angle is called the phase, and turning a qubit means changing it.
- Square the length to get a probability.
- Minus signs are the point — they are what lets routes cancel.