Introducing the Qubit

A classical bit holds exactly one of two values — 0 or 1. A qubit is the quantum analogue: a physical two-level system whose state is described by a complex vector rather than a single digit.

Two levels, complex amplitudes

Any isolated two-level quantum system can serve as a qubit: the spin of an electron, the polarisation of a photon, or the energy levels of a trapped ion. What matters is that exactly two orthogonal basis states exist. Following Dirac notation we call them $|0\rangle$ and $|1\rangle$ (read "ket zero" and "ket one"), represented as column vectors

The most general pure state of a qubit is a superposition

subject to the normalisation constraint $|\alpha|^2 + |\beta|^2 = 1$. The numbers $\alpha$ and $\beta$ are probability amplitudes: when the qubit is measured, the Born rule assigns probability $|\alpha|^2$ to outcome 0 and $|\beta|^2$ to outcome 1.

The computational basis

The two special states $|0\rangle$ and $|1\rangle$ form the computational basis. They are the quantum analogues of the classical bit values 0 and 1, and they are the only states that give a certain, repeatable measurement outcome. Any other single-qubit pure state has $|\alpha| > 0$ and $|\beta| > 0$ simultaneously, meaning neither outcome is certain before measurement.

Try it

Our simulator initialises every qubit in $|0\rangle$. Apply the Pauli-X gate — the quantum equivalent of a classical NOT — to flip the qubit from $|0\rangle$ to $|1\rangle$. Verify that the resulting state vector matches $|1\rangle = (0, 1)^T$.

After running the circuit, the amplitude of $|0\rangle$ drops to 0 and the amplitude of $|1\rangle$ rises to 1. The X gate is the simplest evidence that a qubit obeys the same linear rules as a full quantum state vector — even when the state stays in the computational basis.

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