Amplitudes, not just probabilities
A regular bit is either 0 or 1. A qubit can be in a [[superposition|superposition]] of both — but the word 'superposition' gets thrown around so loosely that it's worth slowing down. A qubit's state is written with two numbers, called amplitudes: one attached to the outcome 0, and one attached to the outcome 1. We usually call them alpha and beta.
|psi> = alpha|0> + beta|1>
Here's the part that makes quantum different from ordinary probability: amplitudes are not probabilities. A probability is always a number between 0 and 1. An amplitude can be negative, and in general it's a complex number (it carries a kind of direction, or phase). That sounds like a technicality, but it is the whole story — negative and complex amplitudes are exactly what lets quantum states do something a coin flip never could.
The myth of 'all answers at once'
You have probably heard that a quantum computer 'tries all the answers at the same time.' It is the single most common thing said about quantum computing, and it is misleading. Yes, a register of qubits in superposition can hold amplitudes for an enormous number of possibilities at once. But holding amplitudes is not the same as *knowing the answers*.
The catch is what happens when you look. A [[quantum-measurement|measurement]] does not hand you the whole list of possibilities — it returns exactly one outcome, chosen at random with a probability set by the amplitudes. Everything else collapses away. If you simply put a register into an even superposition of a billion possibilities and measured it, you would get one random possibility, no better than guessing. The superposition by itself buys you nothing.
Interference: the real trick
This is where those negative and complex amplitudes finally earn their keep. Because amplitudes can be positive or negative, they can interfere — just like ripples on a pond. When two paths to the same outcome have amplitudes with the same sign, they add up and reinforce (constructive interference). When they have opposite signs, they cancel (destructive interference). This is [[quantum-interference|quantum interference]], and it is the actual engine of quantum computing.
A real quantum algorithm is a careful choreography of amplitudes. You arrange the computation so that the amplitudes leading to wrong answers cancel out toward zero, while the amplitudes leading to the right answer add up. Only then do you measure. Because the wrong outcomes now have tiny squared amplitudes, you are very likely to read out the answer you wanted. No parallel-universe brute force — just interference, engineered on purpose.
Normalization
There is one rule the amplitudes must always obey. Since a measurement gives some outcome, the probabilities of all outcomes have to add to 1. For a single qubit that means the squared sizes of the two amplitudes must sum to one: |alpha|^2 + |beta|^2 = 1. This is called normalization, and a valid quantum state is always normalized.
Normalization also quietly enforces a trade-off. If a qubit is split evenly, alpha and beta each have squared size 1/2, so you have a 50/50 chance of either outcome. Push more amplitude toward 0 and you necessarily pull it away from 1. You can move certainty around, but you can never have more than a total of 1 to spend.
Why you can't just read it all out
It's tempting to think you could prepare a rich superposition, measure it, and somehow recover all the amplitudes hidden inside. You can't. A single [[quantum-measurement|measurement]] returns one outcome — 0 or 1 for a qubit — and the moment it does, the superposition is gone (it 'collapses'). You never get to see alpha and beta directly. You only ever get a sample drawn from |alpha|^2 and |beta|^2.
Could you make a backup copy of the state before measuring, to learn more? No — quantum mechanics forbids copying an unknown state outright, a rule with real consequences we'll meet later. And running the same computation many times only gives you statistics about the probabilities; it still never reveals the underlying amplitudes or their phases directly.