Quantum Bits (Qubits)

Mathematically, a qubit is a unit vector in a two-dimensional complex Hilbert space. We write its state in Dirac notation as

\[|\psi\rangle = \alpha|0\rangle + \beta|1\rangle, \qquad \alpha, \beta \in \mathbb{C}, \quad |\alpha|^2 + |\beta|^2 = 1.\]

The complex numbers \(\alpha\) and \(\beta\) are probability amplitudes. The Born rule says: measure the qubit in the computational basis and you observe \(|0\rangle\) with probability \(|\alpha|^2\) and \(|1\rangle\) with probability \(|\beta|^2\). After measurement the state collapses to whichever outcome you saw — the superposition is destroyed.

Up to an irrelevant global phase, every single-qubit state can be written as

\[|\psi\rangle = \cos\frac{\theta}{2}\,|0\rangle + e^{i\phi}\sin\frac{\theta}{2}\,|1\rangle,\]

which maps one-to-one onto a point \((\theta, \phi)\) on the Bloch sphere: \(|0\rangle\) at the north pole, \(|1\rangle\) at the south pole, and every equal superposition — e.g. \(|+\rangle = \tfrac{|0\rangle+|1\rangle}{\sqrt{2}}\) — on the equator.

Qubit vs. probabilistic bit. A classical coin flip is also “0 or 1 with probabilities” — but its randomness is ignorance: the coin already landed, you just don't know how. A qubit's superposition is ontological: before measurement there is no fact of the matter, and the amplitudes \(\alpha, \beta\) can interfere. No classical probability distribution reproduces interference — the later lessons prove exactly this.

For \(n\) qubits the state lives in a \(2^n\)-dimensional space — the tensor product of the single-qubit spaces. Writing down the state of just 50 qubits already needs more numbers than any classical supercomputer can store. That is the origin of quantum advantage — and of why simulating quantum systems is exponentially hard classically.