Background Quantum Mechanics

A qubit is normally represented as a linear combination of the basis states $\vert\rangle$ and $\vert 1\rangle$, ie;

$$\vert\phi\rangle = \alpha\vert\rangle + \beta\vert 1\rangle$$ (1)

It is a fundamental property of quantum mechanics that we cannot examine a qubit to determine it's quantum state, in other words, to find out what $\alpha$ and $\beta$ are. Instead, upon measurement, a qubit collapses into the state $\vert\rangle$ with probability $\vert\alpha\vert^2$ and the state $\vert 1\rangle$ with probability $\vert\beta\vert^2$, such that;

$$\vert\alpha\vert^2 + \vert\beta\vert^2 = 1$$

Because of this requirement, equation (1) can be rewritten as;

$$\vert\phi\rangle = e^{i\gamma}\left(cos\frac{\theta}{2}\vert\rangle + e^{i\varphi} sin\frac{\theta}{2}\vert 1\rangle \right),$$

where $\theta,\ \varphi\ and\ \gamma$ are real numbers. We can ignore the factor of $e^{i\gamma}$ at the front of the equation, because it has no observable effects, and we can therefore rewrite equation (1) as;

$$\vert\phi\rangle = cos\frac{\theta}{2}\vert\rangle + e^{i\varphi} sin\frac{\theta}{2}\vert 1\rangle$$ (2)