Quantum computing, from one qubit up.

A qubit is a two-level quantum system. Its state is a unit vector in a 2-dimensional complex space, written as a combination of the basis states |0⟩ and |1⟩:

α and β are complex amplitudes. They are not probabilities themselves — their squared magnitudes are.

When both α and β are non-zero, the qubit is in a superposition. This is not "secretly 0 or 1" — the amplitudes can interfere, which is impossible for a classical coin with hidden values.

Measuring in the computational basis yields 0 with probability |α|² and 1 with probability |β|² (the Born rule). Afterwards the state collapses to the observed basis state.

Gates are unitary matrices — reversible operations that preserve total probability. A few essentials:

Ignoring an unobservable global phase, any pure qubit state can be written with two angles and drawn as a point on a unit sphere — the Bloch sphere. Single-qubit gates are rotations of this sphere.

Two qubits are entangled when their joint state cannot be written as a product of single-qubit states. Measurement outcomes are then correlated beyond anything classical shared randomness can explain (Bell's theorem).

Amplitudes add like waves. Paths with opposite signs cancel (destructive interference) and with equal signs reinforce. Quantum algorithms are choreographed interference.

A quantum circuit lists gates applied to qubit wires over time, read left to right, usually ending in measurement. For n qubits the simulator tracks 2ⁿ complex amplitudes — which is why classical simulation becomes infeasible beyond roughly 50 qubits.

Grover's search finds a marked item among N in about √N steps. Shor's algorithm factors integers using the quantum Fourier transform. Both rely on interference to boost the right answers.

Quantum computers are not universally faster. They offer speed-ups for specific problems — simulation of quantum systems, certain search and factoring tasks — while today's hardware is small and noisy. A qubit is not "both 0 and 1 at once, so computers try every answer": measurement returns one outcome, and useful algorithms must engineer interference.