Classical Bits vs Qubits
A bit holds one readable value, 0 or 1. A qubit carries two amplitudes — one for each outcome — and measuring it returns a single bit, with probability equal to each amplitude's squared magnitude.
What is a bit, and what is a qubit?
A classical bit holds one value, 0 or 1. A program can read it, copy it, and branch on it at any time.
A qubit is a physical system with two reliably distinguishable configurations, labelled |0⟩ and |1⟩ (the bracket notation simply names a quantum state). Its state is described by two numbers called amplitudes: α attached to |0⟩ and β attached to |1⟩, written α|0⟩ + β|1⟩ (the + does not add anything here — the expression simply lists the two outcomes side by side, each with its amplitude attached) (the + does not add anything here — the expression simply lists the two outcomes side by side, each with its amplitude attached).
Amplitudes can be negative, or even complex numbers. They are internal bookkeeping — you never read them directly off a single qubit.
How do amplitudes turn into readout?
Measurement converts the state into one classical bit: you get 0 with probability α squared, and 1 with probability β squared (for complex amplitudes, square the magnitude). This rule — probability equals squared amplitude — is called the Born rule.
Worked example. Take α = 0.6 and β = 0.8. Then 0.6 × 0.6 = 0.36 and 0.8 × 0.8 = 0.64, so 36% of measurements read 0 and 64% read 1. The check: 0.36 + 0.64 = 1.00 exactly — a valid state's probabilities always sum to 1. That requirement is called normalisation.
Second worked example. The pair (1, 1) is not a valid state: 1 + 1 = 2, not 1. Dividing both amplitudes by √2 ≈ 1.4142 repairs it: each becomes 0.7071, each square is 0.5, and 0.5 + 0.5 = 1.
Which one-qubit states should you know by name?
- |0⟩ — amplitudes (1, 0). Always reads 0.
- |1⟩ — amplitudes (0, 1). Always reads 1.
- |+⟩ ("plus") — amplitudes (0.7071, 0.7071). Reads 0 or 1, 50/50.
- |−⟩ ("minus") — amplitudes (0.7071, −0.7071). Also reads 0 or 1, 50/50.
The last two differ only in the sign of one amplitude. A sign (or complex-factor) difference between amplitudes is called relative phase. Squaring erases it — (−0.7071) × (−0.7071) = 0.5, the same as for +0.7071 — so |+⟩ and |−⟩ produce identical histograms here.
Identical histograms do not mean identical states. Later gates can convert that hidden sign into different probabilities — the seed of every quantum algorithm.
Run it: prepare |+⟩
What does a qubit not give you?
- It does not store two readable classical values at once.
- One measurement does not reveal α and β; it returns a single bit.
- You cannot copy an unknown qubit state the way you copy a variable — quantum mechanics forbids a machine that duplicates an arbitrary unknown state (the no-cloning theorem).
- A simulator's state-vector panel is a privileged debugging view of the mathematics, not an operation any hardware offers on an unknown state.
Is a qubit "both 0 and 1"?
The slogan hides the machinery. A qubit has one definite quantum state; that state assigns amplitudes to the outcomes 0 and 1; a specified measurement returns exactly one of them.
Saying "the amplitudes for 0 and for 1 are both non-zero" is longer but true — and it points at the two numbers you can actually calculate with.
Developer model: treat the state as private internal data with a restricted read API. Gates update the amplitudes; measurement samples a bit. The restriction is part of the computational model, not an inconvenience of current hardware.