mirror of
https://github.com/Steffo99/appunti-magistrali.git
synced 2024-11-25 03:34:17 +00:00
30 lines
1 KiB
Markdown
30 lines
1 KiB
Markdown
Un [[gate quantistico]] che permette di effettuare una rotazione su asse arbitrario.
|
|
|
|
Usando la [[formula di Eulero]], esso corrisponde a:
|
|
$$
|
|
\def \varX {{\color{coral} \theta}}
|
|
\def \varY {{\color{cornflowerblue} \phi}}
|
|
\def \varZ {{\color{yellowgreen} \lambda}}
|
|
\def \varI {{\color{hotpink} i}}
|
|
\Huge
|
|
\mathbf{U}(\varX, \varY, \varZ) = \begin{bmatrix}
|
|
\cos \left( \frac{\varX}{2} \right) &
|
|
- e^{\varI \varZ} \sin \left( \frac{\varX}{2} \right) \\
|
|
e^{\varI \varY} \sin \left( \frac{\varX}{2} \right) &
|
|
e^{\varI \varY + \varI \varZ} \sin \left( \frac{\varX}{2} \right)
|
|
\end{bmatrix}
|
|
$$
|
|
|
|
Espanso, sarebbe:
|
|
$$
|
|
\def \varX {{\color{coral} \theta}}
|
|
\def \varY {{\color{cornflowerblue} \phi}}
|
|
\def \varZ {{\color{yellowgreen} \lambda}}
|
|
\def \varI {{\color{hotpink} i}}
|
|
\mathbf{U}(\varX, \varY, \varZ) = \begin{bmatrix}
|
|
\cos \frac{\varX}{2} &
|
|
- (\cos \varZ + \varI \sin \varZ) \cdot \sin \frac{\varX}{2} \\
|
|
(\cos \varY + \varI \sin \varY) \cdot \sin \frac{\varX}{2} &
|
|
(cos (\varY + \varZ) + \varI \sin (\varY + \varZ)) \cdot \sin \frac{\varX}{2}
|
|
\end{bmatrix}
|
|
$$
|