Trigonometry

Summary

Quantum phases and rotations rely on sine/cosine. Unit complex numbers sit on the unit circle: $$e^{i\theta} = \cos\theta + i \sin\theta$$ Small rotations compound.

SU(2) rotations: $$R(\theta) = \begin{bmatrix}\cos\theta & -\sin\theta \\ \sin\theta & \cos\theta\end{bmatrix}$$
Angles in radians are natural for gates
Phase shifts add: $$e^{i\alpha}e^{i\beta} = e^{i(\alpha+\beta)}$$

Practice

1) If $e^{i\theta} = \cos\theta + i\sin\theta$, what is $e^{-i\theta}$?

$\cos\theta - i\sin\theta$ is the complex conjugate on the unit circle.

2) Compute $\cos(\alpha+\beta)$ in terms of $cos$/$sin$.

$\cos\alpha \cos\beta - \sin\alpha \sin\beta$.

3) For small $\theta$, approximate $\sin\theta$ and $\cos\theta$ to first order.

$\sin\theta \approx \theta$, $\cos\theta \approx 1 - \tfrac{\theta^2}{2}$.

Trigonometry

Summary

Practice

1) If \(e^{i\theta} = \cos\theta + i\sin\theta\), what is \(e^{-i\theta}\)?

2) Compute \(\cos(\alpha+\beta)\) in terms of \(cos\)/\(sin\).

3) For small \(\theta\), approximate \(\sin\theta\) and \(\cos\theta\) to first order.