qubits

A qubit is a vector in a two-dimensional quantum system.

What is a two-dimensional quantum system? It's a complex vector space where the basis contains two quantum states. What's a quantum state? It's a vector in a vector space.

\[\alpha \ket{0} + \beta\ket{1}\] where \(\alpha\) and \(\beta\) are complex numbers such that \(|\alpha|^2 + |\beta|^2 = 1\)

This can also be written as \[e^{i\gamma} \left(\cos\frac{\theta}{2}\ket{0} + e^{i\phi}\sin\frac{\theta}{2}\ket{1} \right)\]

Why? If we didn't have the \(\sin\) and \(\cos\), we would have \[e^{i\gamma}\ket{0} + e^{i(\phi+\gamma)}\ket{1}\]

We see that each coefficient can be an arbitrary complex number with magnitude 1. Now recall that \(\sin^2\theta + \cos^2\theta = 1\). So we multiply by the \(\sin\) and \(\cos\) terms to scale the magnitudes so they add up to 1.

See Mike and Ike for the formula.