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###### tags: `Quantum Computing`
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# Bloch Sphere
## def
- magnitude=amplitude
- phase: $e^{i\theta}$
- Global Phase: any complex $\alpha, \beta$
- Relative Phase:0~1
- $|\alpha|=\cos(\theta/2)$, $|\beta|=\sin(\theta/2)$

## phase: $e^{i\theta}$
$|\psi> =\alpha|0>+ \beta|1>$
$\because \alpha, \beta$ $\in Complex$

$\therefore$ it can be rewrited as $|\psi> =|\alpha|e^{i\theta_\alpha}|0>+ |\beta|e^{i\theta_\beta}|1>$
$|\psi> =[|\alpha||0>+ |\beta|e^{i(\theta_\beta-\theta_\alpha)}|1>]$
$e^{i\theta_\alpha}$ is **global phase**
$e^{i(\theta_\beta-\theta_\alpha)}$ is **relative phase**(between 0 and 1)

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## bloch sphere


$|\alpha|^2+|\beta|^2=1$
$\cos^2(\theta/2)+\sin^2(\theta/2)=1$
$|\alpha|=\cos(\theta/2)$
$|\beta|=\sin(\theta/2)$

=> $|\psi >=\cos(\theta/2)|0>+\sin(\theta/2)e^{i\phi} |1>$

$e^{i\pi}=-1, \because \cos\theta=-1且\sin\theta=0$